Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 83, pp. 1–110.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CUBIC DIFFERENTIAL SYSTEMS WITH INVARIANT

STRAIGHT LINES OF TOTAL MULTIPLICITY SEVEN AND

FOUR REAL DISTINCT INFINITE SINGULARITIES

CRISTINA BUJAC, DANA SCHLOMIUK, NICOLAE VULPE

Abstract. In this article we consider the class CSL4s∞
7 of non-degenerate

real planar cubic vector fields possessing four distinct real infinite singularities
and invariant straight lines, including the line at infinity of total multiplicity 7.

We prove that there are exactly 93 distinct configurations of invariant straight

lines for this class, and present corresponding examples for the realization of
each one of the detected configurations.

1. Introduction and statement of main results

In this article we consider the real polynomial differential system

dx

dt
= P (x, y),

dy

dt
= Q(x, y), (1.1)

where P and Q are polynomials in x, y with real coefficients, i.e. P,Q ∈ R[x, y].
The degree of a system is defined as max(deg(P ),deg(Q)), so that a cubic system
is a system of degree three.

We also consider the vector field corresponding to (1.1):

X = P (x, y)
∂

∂x
+Q(x, y)

∂

∂y

Darboux [12] introduced the notion of an algebraic invariant curve for differential
equations on the complex plane. An algebraic curve f(x, y) = 0 with f(x, y) ∈
C[x, y] is an invariant curve of a system of the form (1.1) where P (x, y), Q(x, y) ∈
C[x, y] if and only if there exists K[x, y] ∈ C[x, y] such that

X(f) = P (x, y)
∂f

∂x
+Q(x, y)

∂f

∂y
= f(x, y)K(x, y)

is an identity in C[x, y]. Since R ⊂ C, any system (1.1) over R generates a system
of differential equation over C. Using the embedding C2 ↪→ P2(C), (x, y) 7→ [x :
y : 1] = [X : Y : Z], (x = X/Z, y = Y/Z and Z 6= 0), we can compactify
the differential equation Q(x, y)dy − P (x, y)dx = 0 to an associated differential

2010 Mathematics Subject Classification. 58K45, 34C05, 34A34.
Key words and phrases. Cubic vector field; invariant straight line; multiplicity of singularity;

infinite and finite singularities; multiplicity of invariant lines;

configurations of invariant straight lines.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted December 21, 2018. Published October 3, 2021.

1



2 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

equation over the complex projective plane. In fact the theory of Darboux in [12]
is done for differential equations on the complex projective plane.

We compactify the space of all the polynomial differential system (1.1) of degree
n on SN−1 with N = (n+ 1)(n+ 2) by multiplying the coefficients of each system
with 1/(

∑
(a2ij + b2ij))

1/2, where aij and bij are the coefficients of the polynomials
P (x, y) and Q(x, y), respectively.

Definition 1.1 ([31]). (1) We say that an invariant curve L : f(x, y) = 0, f ∈
C[x, y] for a polynomial system (S) of degree n has multiplicity m if there exists
a sequence of real polynomial system (Sk) of degree n converging to (S) in the
topology of SN−1, N = (n+ 1)(n+ 2), such that each (Sk) has m distinct invariant
curves L1,k : f1,k(x, y) = 0, . . . ,Lm,k : fm,k(x, y) = 0 over C, deg(f) = deg(fi,k) =
r, converging to L as k →∞, in the topology of PR−1(C), with R = (r+1)(r+2)/2
and this does not occur for m+ 1.

(2) We say that the line at infinity L∞ : Z = 0 of a polynomial system (S) of
degree n has multiplicity m if there exists a sequence of real polynomial system (Sk)
of degree n converging to (S) in the topology of SN−1, N = (n + 1)(n + 2), such
that each (Sk) has m − 1 distinct invariant lines L1,k : f1,k(x, y) = 0, . . . ,Lm,k :
fm−1,k(x, y) = 0 over C, converging to the line at infinity L∞ as k → ∞, in the
topology of P2(C) and this does not occur for m.

In this work we consider a particular case of invariant algebraic curves, namely
the invariant straight lines of system (1.1). A straight line over C is the locus
{(x, y) ∈ C2|f(x, y) = 0} of an equation f(x, y) = ux + vy + w = 0 with (u, v) 6=
(0, 0) and (u, v, w) ∈ C3. We note that by multiplying the equation by a non-zero
complex number λ, the locus of the equation does not change. So that we have
a bijection between the lines in C2 and the points in P2(C)\{[0 : 0 : 1]}. This
bijection induces a topology on the set of lines in C2 from the topology of P2(C)
and hence we can talk about a sequence of lines convergent to a line in C2.

For an invariant line f(x, y) = ux + vy + w = 0 we denote â = (u, v, w) ∈ C3

and by [â] = [u : v : w] the corresponding point in P2(C). We say that a sequence
of straight lines fi(x, y) = 0 converges to a straight line f(x, y) = 0 if and only if
the sequence of points [âi] converges to [â] = [u : v : w] in the topology of P2(C).

In view of the above definition of an invariant algebraic curve of a system (1.1),
a line f(x, y) = ux + vy + w = 0 over C is an invariant line if and only if it there
exists K(x, y) ∈ C[x, y] which satisfies the following identity in C[x, y]:

X(f) = uP (x, y) + vQ(x, y) = (ux+ vy + w)K(x, y).

We point out that if we have an invariant line f(x, y) = 0 over C it could happen
that multiplying the equation by a number λ ∈ C∗ = C\{0}, the coefficients of the
new equation becomes real, i.e. (uλ, vλ,wλ) ∈ R3. In this case, along with the line
f(x, y) = 0 sitting in C2 we also have an associated real line, sitting in R2 defined
by λf(x, y) = 0.

Note that, since a system (1.1) is with real coefficients, if its associated complex
system has a complex invariant straight line ux + vy + w = 0, then its conjugate
complex invariant straight line ūx+ v̄y + w̄ = 0 is also invariant.

A line in P2(C) is the locus in P2(C) of an equation F (X,Y, Z) = uX+vY +wZ =
0 where (u, v, w) ∈ C3 and F (X,Y, Z) ∈ C[X,Y, Z]. The line Z = 0 in P2(C) is
called the line at infinity of the affine plane C2. This line is an invariant manifold
of the complex differential equation on P2(C. Clearly the lines in P2(C) are in a



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 3

one-to-one correspondence with points [u : v : w] ∈ P2(C) and thus we have a
topology on the set of lines in P2(C). We can thus talk about a sequence of lines
in P2(C) convergent to a line in P2(C).

To a line f(x, y) = ux + vy + w = 0, (u, v) 6= (0, 0), f ∈ C[x, y], we associate
its projective completion F (X,Y, Z) = uX + vY + wZ = 0 under the embedding
C2 ↪→ P2(C), (x, y) 7→ [x : y : 1] = [X,Y, Z] indicated above.
We first remark that in the above definition we made an abuse of language. Indeed,
we talk about complex invariant lines of real system. However we already said that
to a real system one can associate a complex system and to a differential equation
Q(x, y)dy − P (x, y)dx = 0 corresponds a differential equation in P2(C).

We remark that the above definition is a particular case of the definition of
geometric multiplicity given in [11], and namely the ”strong geometric multiplicity”
with the restriction, that the corresponding perturbations are cubic system.

The set CS of cubic differential system depends on 20 parameters and for this
reason people began by studying particular subclasses of CS. Some of these sub-
classes are on cubic system having invariant straight lines.

We mention here some papers on polynomial differential system possessing in-
variant straight lines. For quadratic system see [13, 27, 26, 31, 32, 33, 34, 35] and
[36]; for cubic system see [18, 21, 22, 20, 28, 39, 40, 4, 5, 6, 7, 8, 9, 10] and [29]; for
quartic system see [38] and [42].

The existence of sufficiently many invariant straight lines of planar polynomial
system could be used for proving the integrability of such system. During the past
15 years several articles were published on this theme (see for example [32, 34]).

According to [1], for a non-degenerate polynomial differential system of degree
m, the maximum number of invariant straight lines including the line at infinity
and taking into account their multiplicities is 3m. This bound is always reached
(see [11]).

In particular, the maximum number of the invariant straight lines (including the
line at infinity Z = 0) for cubic systems with a finite number of infinite singularities
is 9. In [20] the authors classified all cubic systems possessing the maximum number
of invariant straight lines taking into account their multiplicities according to their
configurations of invariant lines. The notion of configuration of invariant lines for
a polynomial differential system was first introduced in [31].

Definition 1.2 ([35]). Consider a real planar polynomial differential system (1.1).
We call configuration of invariant straight lines of this system, the set of (complex)
invariant straight lines (which may have real coefficients) including the line at in-
finity of the system, each endowed with its own multiplicity and together with all
the real singular points of this system located on these invariant straight lines, each
one endowed with its own multiplicity.

In [20] the authors used a weaker notion, not taking into account the multiplic-
ities of real singularities. They detected 23 such configurations. Moreover, in [20]
the necessary and sufficient conditions for the realization of each one of 23 config-
urations detected, are determined using invariant polynomials with respect to the
action of the group of affine transformations (Aff(2,R)) and time rescaling (i.e.
Aff(2,R) × R∗)). In [4] the author detected another class of cubic system whose
configuration of invariant lines was not detected in [20].

If two polynomial systems are equivalent under the action of the affine group and
time rescaling, clearly they must have the same kinds of configurations of invariant



4 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

lines. But it could happen that two distinct polynomial systems which are non-
equivalent modulo the action of the affine group and time rescaling have “the same
kind of configurations” of straight lines. We need to say when two configurations
are considered equivalent.

Definition 1.3. Suppose we have two cubic systems (S), (S′) both with a finite
number of singularities, finite and infinite, a finite set of invariant straight lines
Li : fi(x, y) = 0, i = 1, . . . , k, of (S) (respectively L′i : f ′i(x, y) = 0, i = 1, . . . , k′, of
(S′)). We say that the two configurations C,C ′ of invariant lines, including the line
at infinity, of these systems are equivalent if there is a one-to-one correspondence
φ between the lines of C and C ′ such that:

(i) φ sends an affine line (real or complex) to an affine line and the line at
infinity to the line at infinity conserving the multiplicities of the lines and also
sends a invariant line with coefficients in R to an invariant line with coefficients in
R;

(ii) for each line L : f(x, y) = 0 we have a one-to-one correspondence between
the real singular points on L and the real singular points on φ(L) conserving their
multiplicities and their order on these lines;

(iii) we have a one-to-one correspondence φ∞ between the real singular points
at infinity on the (real) lines at infinity of (S) and (S′) such that when we list in
a counterclock wise sense the real singular points at infinity on (S) starting from a
point p on the Poincaré disk, p1 = p,. . . ,pk, φ∞ preserves the multiplicities of the
singular points and preserves or reverses the orientation;

(iv) consider the total curves

F :
∏

Fj(X,Y, Z)miZm = 0,F ′ :
∏

F ′j(X,Y, Z)m
′
iZm = 0

where Fi(X,Y, Z) = 0 (respectively F ′i (X,Y, Z) = 0) are the projective completions
of Li (respectively L′i) and mi,m

′
i are the multiplicities of the curves Fi = 0, F ′i = 0

and m,m′ are respectively the multiplicities of Z = 0 in the first and in the second
system. Then, there is a one-to-one correspondence ψ between the real singularities
of the curves F and F ′ conserving their multiplicities as singular points of the total
curves.

Remark 1.4. To describe the various kinds of multiplicity for infinite singular
points we use the concepts and notations introduced in [31]. Thus we denote by
“(a, b)” the maximum number a (respectively b) of infinite (respectively finite)
singularities which can be obtained by perturbation of a multiple infinite singular
point.

The configurations of invariant straight lines which were detected for some fam-
ilies of system (1.1), were instrumental for determining the phase portraits of those
families. For example, in [32, 34] it was proved that we have a total of 57 distinct
configurations of invariant lines for quadratic system with invariant lines total mul-
tiplicity greater than or equal to 4. These 57 configurations lead to the existence of
135 topologically distinct phase portraits. In [28, 39, 40, 29] it was proved that cubic
system with invariant lines of total parallel multiplicity six or seven (the notion of
”parallel multiplicity” could be found in [40]) have 113 topologically distinct phase
portraits. This was done by using the various possible configurations of invariant
lines of these system.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 5

In what follows we define some algebraic-geometric notions which will be needed
in order to describe the invariants used for distinguishing configurations and phase
portraits.

Let V be an irreducible algebraic variety of dimension n over a field K.

Definition 1.5. A cycle of dimension r or r-cycle on V with coefficients in an
Abelian group G is a formal sum ΣWnWW , where W is a subvariety of V of
dimension r which is not contained in the singular locus of V , nW ∈ G, and
only a finite number of nW are non-zero. The support of a cycle C is the set
Supp(C) = {W |nW 6= 0}. An (n− 1)-cycle is called a divisor D.

Definition 1.6. We call type of a divisor D the set of all ordered couples (m, sm)
where m is an integer appearing as a coefficient in the divisor D and sm is the
number of occurrences in D of the coefficient m.

Clearly the notion of type of a divisor is an affine invariant.
These notions (see [16]) which occur frequently in algebraic geometry, were

used for classification purposes of planar quadratic differential system by Pal and
Schlomiuk [24], [30] and by Llibre and Schlomiuk in [19]. They are also helpful here
as we indicate below.

We apply the preceding notions to planar polynomial differential system (1.1).
We denote by PSLn,L the class of all non-degenerate planar polynomial differential
system of degree n with a finite number of infinite singularities and possessing
invariant lines, including the line at infinity, of total multiplicity L.

We define here below an important divisor which is used in this work and which
we call the parallelism divisor. Consider a system in (S) ∈ PSLn,L. Let p1, p2, . . . , ps
be the set of all the real singular points at infinity of (S). Let jk, k ∈ {1, . . . , s}
be the total multiplicity of all invariant affine lines which cut the line at infinity
at pk. Let ik, k ∈ {1, . . . , s} be the maximum number of distinct invariant affine
lines which can appear from the line at infinity in a perturbation of (S) in the class
PSLn,L and which cut the line at infinity at pk.

Definition 1.7. We call parallelism divisor on Z = 0 with coefficients in Z2 the
divisor DL(S;Z) defined as follows:

DL(S;Z) =

s∑
k=1

(
ik
jk

)
pk.

In this definition we spell out the affine part jk (the finite parallelism index)
as well as the infinite part expressed by ik (the infinite parallelism index). We
could form another divisor on the line at infinity, namely

∑s
k=1 (ik + jk)pk whose

coefficients are the total parallelism indices.

Definition 1.8. We define the parallelism type of the configuration (or simply type
of the configuration) of invariant lines occurring for a cubic polynomial system (S),
the sequence of non-zero numbers, τk = ik+jk, k ∈ {1, . . . , s} attached to DL(S;Z),
listed according to descending magnitudes:

T = (τ1, τ2, . . . , τl), 1 ≤ l ≤ s.

Clearly T is an affine invariant of system in the class PSLn,L and of their con-
figurations of invariant lines.



6 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Notation 1.9. We shall denote by CSL4s∞
7 the class of cubic system with invariant

lines of total multiplicity seven which have four distinct real singularities at infinity.

In this article we classify the family CSL4s∞
7 according to the relation of equiv-

alence of configurations. Our main result is the following one.

Theorem 1.10. The class CSL4s∞
7 has a total of 93 non-equivalent configurations

of invariant lines, only 20 of which have complex invariant lines and these are
always simple, 10 of which are with only one couple of complex conjugate invariant
lines and 10 with two couples of complex conjugate invariant lines. The remaining
74 configurations have only invariant lines whose coefficients could be made real,
of multiplicities at most three and in anyone of the configurations, there is at most
one line of multiplicity three. There is a total of five configurations with a triple
invariant line, in two of them this being the line at infinity. The 93 configurations
Config. 7.1 – Config. 7.93 of invariant straight lines are given in Figure 1. The
configurations split into subclasses according to the value of the invariant T as
follows:

• 14 configurations Config. 7.1 – Config. 7.14 are of the type T = (3, 3);
• 26 configurations Config. 7.15 – Config. 7.40 are of the type T = (3, 2, 1);
• 25 configurations Config. 7.41 – Config. 7.65 are of the type T = (3, 1, 1, 1);
• 1 configuration Config. 7.66 is of the type T = (2, 2, 2);
• 27 configurations Config. 7.67 – Config. 7.93 are of the type T = (2, 2, 1, 1).

We prove that each one of these configurations is realizable within CSL4s∞
7 by

constructing examples for each one of the configurations Config. 7.1–Config. 7.93.
The proof that all these 93 configurations are non-equivalent, according to our
definition of equivalence is done in Subsection 3.6.

Notation 1.11. We explain here how to read the pictures representing the config-
urations. An invariant line with multiplicity k > 1 will appear in a configuration
in bold face and will have next to it the number k. Real invariant straight lines
are represented by continuous lines, whereas complex invariant straight lines are
represented by dashed lines. The multiplicities of the real singular points of the
system located on the invariant lines, will be indicated next to the singular points.
The maximum number of parallel invariant straight lines will be shown to be three.
Whenever we have three parallel lines, clearly at least for one of these will be real.
Due to an affine transformation we can assume this line to be x = 0 and after this
transformation the system will be of the form:

ẋ = x(a+ 2bx+ cx2), ẏ = Q(â, x, y).

Here Q(â, x, y) = a0 + a10x + a01y + a20x
2 + 2a11xy + a02y

2 + a30x
3 + 3a21x

2y +
3a12xy

2 + a03y
3 and â = (a0, a10, . . . , a03). If two invariant lines of the triplet are

complex, then the condition b2 − ac < 0 must hold and this implies that c 6= 0 and
due to time rescaling we may assume c = 1. Setting b2− a = −u2 (a = b2 + u2) we
obtain the system

ẋ = x[(x+ b)2 + u2],

ẏ = Q(â, x, y).
(1.2)

which has the triplet of invariant lines: x = 0, x = −b + iu, x = −b − iu. In case
b 6= 0 we place both complex invariant lines on one side of the real line. If b = 0 we
make the convention to place this line between the two complex lines.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 7

Figure 1. Configurations of invariant lines of total multiplicity 7
for cubic system with 4 distinct real infinite singularities (to be
continued)

The work is organized as follows. In Section 2 we give some preliminary results
needed for this paper. In Section 3 we prove our Main Theorem restricting ourselves
to cubic system with exactly four distinct singularities at infinity. In Subsection 3.1–
3.5 we examine step by step each one of the five possible types of the configurations



8 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Figure 1. (cont.) Configurations of invariant lines of total mul-
tiplicity 7 for cubic system with 4 distinct real infinitesingularities
(to be continued)

defined above (see Definition 1.8). In Subsection 3.6, using the geometric invariants,
we prove that all the 93 detected configurations of invariant lines for the class of
cubic system we considered are distinct according to Definition 1.3.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 9

Figure 1. (cont.) Configurations of invariant lines of total multi-
plicity 7 for cubic system with 4 distinct real infinite singularities
(to be continued)

We note that the construction of the affine invariant necessary and sufficient
conditions for the distinction of the configurations as well as for the realization of
each one of them will be the subject of a new article which is in progress.



10 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Figure 1. (cont.) Configurations of invariant lines of total multi-
plicity 7 for cubic system with 4 distinct real infinite singularities
(to be continued)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 11

Figure 1. (cont.) Configurations of invariant lines of total mul-
tiplicity 7 for cubic system with 4 distinct real infinite singularities

2. Preliminaries

Consider real cubic system, i.e. system of the form:

ẋ = p0 + p1(x, y) + p2(x, y) + p3(x, y) ≡ P (a, x, y),

ẏ = q0 + q1(x, y) + q2(x, y) + q3(x, y) ≡ Q(a, x, y)
(2.1)

with variables x and y and real coefficients. The polynomials pi and qi (i = 0, 1, 2, 3)
are homogeneous polynomials of degree i in x and y:

p0 = a00, p3(x, y) = a30x
3 + 3a21x

2y + 3a12xy
2 + a03y

3,

p1(x, y) = a10x+ a01y, p2(x, y) = a20x
2 + 2a11xy + a02y

2,

q0 = b00, q3(x, y) = b30x
3 + 3b21x

2y + 3b12xy
2 + b03y

3,

q1(x, y) = b10x+ b01y, q2(x, y) = b20x
2 + 2b11xy + b02y

2.



12 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Let a = (a00, a10, a01, . . . , a03, b00, b10, b01, . . . , b03) be the 20-tuple of the coefficients
of system (2.1) and denote

R[a, x, y] = R[a00, a10, a01, . . . , a03, b00, b10, b01, . . . , b03, x, y].

It is known that on the set of polynomial systems (1.1), in particular on the set
CS of all cubic differential system (2.1), acts the group Aff(2,R) of affine transfor-
mation on the plane [35]. For every subgroup G ⊆ Aff(2,R) we have an induced
action of G on CS. We can identify the set CS of system (2.1) with a subset of
R20 via the map CS → R20 which associates to each system (2.1) the 20-tuple
a = (a00, a10, a01, . . . , a03, b00, b10, b01, . . . , b03) of its coefficients.

For the definitions of an affine or GL-comitant or invariant as well as for the
definition of a T -comitant and CT -comitant we refer the reader to [31]. Here we
shall only construct the necessary invariant polynomials (T -comitants) which are
needed to detect the existence of invariant lines for the class of cubic system with
four real distinct infinite singularities and with invariant straight lines with total
multiplicity seven, including the line at infinity (with its own multiplicity).

Let us consider the polynomials

Ci(a, x, y) = ypi(a, x, y)− xqi(a, x, y) ∈ R[a, x, y], i = 0, 1, 2, 3,

Di(a, x, y) =
∂

∂x
pi(a, x, y) +

∂

∂y
qi(a, x, y) ∈ R[a, x, y], i = 1, 2, 3.

As it was shown in [37] the polynomials{
C0, C1, C2, C3, D1, D2, D3

}
(2.2)

of degree one in the coefficients of system (2.1) are GL-comitants of these system.

Notation 2.1. Let f, g ∈ R[a, x, y] and

(f, g)(k) =

k∑
h=0

(−1)h
(
k

h

)
∂kf

∂xk−h∂yh
∂kg

∂xh∂yk−h
.

(f, g)(k) ∈ R[a, x, y] is called the transvectant of index k of (f, g) (cf. [15], [23]).

Theorem 2.2 ([41]). Any GL-comitant of system (2.1) can be constructed from
the elements of the set (2.2) by using the operations: +, −, ×, and by applying the
differential operation (f, g)(k).

Applying the translation x = x′+x0, y = y′+y0 to the P (a, x, y) and Q(a, x, y),

we obtain P̃ (ã(a, x0, y0), x′, y′) = P (a, x′ + x0, y
′ + y0) and Q̃(ã(a, x0, y0), x′, y′) =

Q(a, x′ + x0, y
′ + y0). We construct the following polynomials

Ωi(a, x0, y0) ≡ Resx′

(
Ci
(
ã(a, x0, y0), x′, y′

)
, C0

(
ã(a, x0, y0), x′, y′

))
/(y′)i+1,

Ωi(a, x0, y0) ∈ R[a, x0, y0], (i = 1, 2, 3)

and we denote

G̃i(a, x, y) = Ωi(a, x0, y0)
∣∣
{x0=x, y0=y}

∈ R[a, x, y] (i = 1, 2, 3).

Remark 2.3. We note that the polynomials G̃1(a, x, y), G̃2(a, x, y) and G̃3(a, x, y)
are affine comitants of system (2.1) and are homogeneous polynomials in the coef-
ficients a00, . . . , b03 and non-homogeneous in x, y and

dega G̃1 = 3, dega G̃2 = 4, dega G̃3 = 5,



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 13

deg(x,y) G̃1 = 8, deg(x,y) G̃2 = 10, deg(x,y) G̃3 = 12.

Notation 2.4. Let Gi(a,X, Y, Z) (i = 1, 2, 3) be the homogenization of G̃i(a, x, y),
i.e.

G1(a,X, Y, Z) = Z8G̃1(a,X/Z, Y/Z),

G2(a,X, Y, Z) = Z10G̃2(a,X/Z, Y/Z),

G3(a,X, Y, Z) = Z12G̃3(a,X/Z, Y/Z),

H(a,X, Y, Z) = gcd
(
G1(a,X, Y, Z),G2(a,X, Y, Z),G3(a,X, Y, Z)

)
in R[a,X, Y, Z].

The geometrical meaning of these affine comitants is given by the two following
lemmas (see [20]).

Lemma 2.5. The straight line L(x, y) ≡ ux+vy+w = 0, u, v, w ∈ C, (u, v) 6= (0, 0)
is an invariant line for a cubic system (2.1) if and only if the polynomial L(x, y) is

a common factor of the polynomials G̃1(x, y), G̃2(x, y) and G̃3(x, y) over C, i.e.

G̃i(x, y) = (ux+ vy + w)W̃i(x, y) (i = 1, 2, 3),

where W̃i(x, y) ∈ C[x, y].

Lemma 2.6. Consider a cubic system (2.1) and let a ∈ R20 be its 20-tuple of
coefficients.

(1) If L(x, y) ≡ ux + vy + w = 0, u, v, w ∈ C, (u, v) 6= (0, 0) is an invariant

straight line of multiplicity k for a system (2.1) then [L(x, y)]k | gcd(G̃1, G̃2, G̃3) in
C[x, y], i.e. there exist Wi(a, x, y) ∈ C[x, y] (i = 1, 2, 3) such that

G̃i(a, x, y) = (ux+ vy + w)kWi(a, x, y), i = 1, 2, 3.

(2) If the line l∞ : Z = 0 is of multiplicity k > 1 then Zk−1 | gcd(G1,G2,G3), i.e.
we have Zk−1 | H(a, X, Y, Z).

Consider the differential operator L = x ·L2−y ·L1 constructed in [3] and acting
on R[a, x, y], where

L1 = 3a00
∂

∂a10
+ 2a10

∂

∂a20
+ a01

∂

∂a11
+

1

3
a02

∂

∂a12
+

2

3
a11

∂

∂a21
+ a20

∂

∂a30

+ 3b00
∂

∂b10
+ 2b10

∂

∂b20
+ b01

∂

∂b11
+

1

3
b02

∂

∂b12
+

2

3
b11

∂

∂b21
+ b20

∂

∂b30
,

L2 = 3a00
∂

∂a01
+ 2a01

∂

∂a02
+ a10

∂

∂a11
+

1

3
a20

∂

∂a21
+

2

3
a11

∂

∂a12
+ a02

∂

∂a03

+ 3b00
∂

∂b01
+ 2b01

∂

∂b02
+ b10

∂

∂b11
+

1

3
b20

∂

∂b21
+

2

3
b11

∂

∂b12
+ b02

∂

∂b03
.

Using this operator and the affine invariant

µ0 = Resultantx
(
p3(a, x, y), q3(a, x, y)

)
/y9

we construct the polynomials

µi(a, x, y) =
1

i!
L(i)(µ0), i = 1, .., 9,

where L(i)(µ0) = L(L(i−1)(µ0)) and L(0)(µ0) = µ0.



14 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

These polynomials are in fact comitants of system (2.1) with respect to the group
GL(2,R) (see [3]). The polynomial µi(a, x, y), i ∈ {0, 1, . . . , 9} is homogeneous of
degree 6 in the coefficients of system (2.1) and homogeneous of degree i in the
variables x and y. The geometrical meaning of these polynomial is revealed in the
next lemma.

Lemma 2.7 ([2, 3]). Assume that a cubic system (S) with coefficients a ∈ R20

belongs to the family (2.1). Then:
(i) The total multiplicity of all finite singularities of this system equals 9 − k if

and only if for every i ∈ {0, 1, . . . , k− 1} we have µi(a, x, y) = 0 in the ring R[x, y]

and µk(a, x, y) 6= 0. In this case the factorization µk(a, x, y) =

k∏
i=1

(uix− viy) 6= 0

over C indicates the coordinates [vi : ui : 0] of singularities at infinity which in
perturbations generate finite singularities of the system (S). Moreover the number
of distinct factors in this factorization is less than or equal to four (the maximum
number of infinite singularities of a cubic system) and the multiplicity of each one
of the factors uix− viy gives us the number of the finite singularities of the system
(S) which have coalesced with the infinite singular point [vi : ui : 0].

(ii) The point M0(0, 0) is a singular point of multiplicity k (1 ≤ k ≤ 9) for
the cubic system (S) if and only if for every i such that 0 ≤ i ≤ k − 1 we have
µ9−i(a, x, y) = 0 in R[x, y] and µ9−k(a, x, y) 6= 0.

(iii) The system (S) is degenerate (i.e. gcd(p, q) 6= const) if and only if
µi(a, x, y) = 0 in R[x, y] for every i = 0, 1, . . . , 9.

To define the invariant polynomials we need, we first construct the following
comitants of second degree with respect to the coefficients of initial system (2.1):

S1 = (C0, C1)
(1)
, S10 = (C1, C3)

(1)
, S19 = (C2, D3)

(1)
,

S2 = (C0, C2)
(1)
, S11 = (C1, C3)

(2)
, S20 = (C2, D3)

(2)
,

S3 = (C0, D2)
(1)
, S12 = (C1, D3)

(1)
, S21 = (D2, C3)

(1)
,

S4 = (C0, C3)
(1)
, S13 = (C1, D3)

(2)
, S22 = (D2, D3)

(1)
,

S5 = (C0, D3)
(1)
, S14 = (C2, C2)

(2)
, S23 = (C3, C3)

(2)
,

S6 = (C1, C1)
(2)
, S15 = (C2, D2)

(1)
, S24 = (C3, C3)

(4)
,

S7 = (C1, C2)
(1)
, S16 = (C2, C3)

(1)
, S25 = (C3, D3)

(1)
,

S8 = (C1, C2)
(2)
, S17 = (C2, C3)

(2)
, S26 = (C3, D3)

(2)
,

S9 = (C1, D2)
(1)
, S18 = (C2, C3)

(3)
, S27 = (D3, D3)

(2)
.

We shall use here the following invariant polynomials constructed in [20] and [6]
in order to determine the necessary conditions for the existence and the numbers
of triplets and/or couples of parallel invariant straight lines which a cubic system
could have (see Theorem 2.10):

V1(a, x, y) =S23 + 2D2
3, V2(a, x, y) = S26, V3(a, x, y) = 6S25 − 3S23 − 2D2

3,

V4(a, x, y) =C3

[
(C3, S23)

(4)
+ 36 (D3, S26)

(2)
]
,

V5(a, x, y) =6C3(9A5 − 7A6) + 2D3(4T16 − T17)− 3T3(3A1 + 5A2) + 3A2T4



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 15

+ 36T 2
5 − 3T44,

where

A1 = S24/288, A2 = S27/72, A3 =
(

72D1A2 +
(
S22, D2

)(1))
/24,

A5 =
(
S23, C3

)(4)
/27/35, A6 =

(
S26, D3

)(2)
/25/33

are affine invariants, whereas the polynomials

T3 = S23/18, T4 = S25/6, T5 = S26/72, T16 =
(
S23, D3

)(2)
/26/33,

T17 =
(
S26, D3

)(1)
/25/33, T44 =

(
(S23, C3)(1), D3

)(2)
/5/26/33,

are T -comitants of cubic system (2.1) (see [31] for the definition of a T -comitant).
We remark that in the above invariant polynomials we keep the notations introduced
in [6].

To determine the degree of the common factor of the polynomials G̃i(a, x, y) for
i = 1, 2, 3, we shall use the notion of the kth subresultant of two polynomials with
respect to a given indeterminate (see for instance, [17], [23]).

Following [20] we consider two polynomials

f(z) = a0z
n + a1z

n−1 + · · ·+ an, g(z) = b0z
m + b1z

m−1 + · · ·+ bm,

in the variable z of degree n and m, respectively.
We say that the k-th subresultant (see for example, [23]) with respect to variable

z of the two polynomials f(z) and g(z) is the (m+n−2k)×(m+n−2k) determinant

R(k)
z (f, g) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a0 a1 a2 . . . . . . am+n−2k−1
0 a0 a1 . . . . . . am+n−2k−2
0 0 a0 . . . . . . am+n−2k−3
. . . . . . . . . . . . . . . . . . . . . . . .
0 0 b0 . . . . . . bm+n−2k−3
0 b0 b1 . . . . . . bm+n−2k−2
b0 b1 b2 . . . . . . bm+n−2k−1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

 (m− k)− times

 (n− k)− times

(2.3)

in which there are m − k rows of a’s and n − k rows of b’s, and ai = 0 for i > n,
and bj = 0 for j > m.

For k = 0 we obtain the standard resultant of two polynomials. In other words
we can say that the k–th subresultant with respect to the variable z of the two
polynomials f(z) and g(z) can be obtained by deleting the first and the last k rows
and the first and the last k columns from its resultant written in the form (2.3)
when k = 0.

The geometrical meaning of the subresultants is based on the following lemma.

Lemma 2.8 (see [17, 23]). Polynomials f(z) and g(z) have precisely k roots in
common (considering their multiplicities) if and only if the following conditions
hold:

R(0)
z (f, g) = R(1)

z (f, g) = R(2)
z (f, g) = · · · = R(k−1)

z (f, g) = 0 6= R(k)
z (f, g).

For the polynomials in more than one variables it is easy to deduce from Lemma
2.8 the following result.



16 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Lemma 2.9. Two polynomials f̃(x1, x2, . . . , xn) and g̃(x1, x2, . . . , xn) have a com-
mon factor of degree k with respect to the variable xj if and only if the following
conditions are satisfied:

R(0)
xj

(f̃ , g̃) = R(1)
xj

(f̃ , g̃) = R(2)
xj

(f̃ , g̃) = · · · = R(k−1)
xj

(f̃ , g̃) = 0 6= R(k)
xj

(f̃ , g̃),

where R
(i)
xj (f̃ , g̃) = 0 in R[x1, . . . xj−1, xj+1, . . . , xn].

In [20] all the possible configurations of invariant lines are determined in the case,
when the total multiplicity of these line (including the line at infinity) equals nine.
All possible configurations of invariant lines in the case when the total multiplicity of
these line (including the line at infinity) equals eight, are determined in [5, 6, 7, 8, 9].

In the above mentioned articles several lemmas are proved concerning the number
of triplets and/or couples of parallel invariant straight lines which could have a cubic
system. Taking together these lemmas produce the following theorem.

Theorem 2.10. If a cubic system (2.1) possesses a given number of triplets or/and
couples of invariant parallel lines real or/and complex, then the following conditions
are satisfied, respectively:

(i) two triplets imply V1 = V2 = U 1 = 0;
(ii) one triplet and one couple imply V4 = V5 = U2 = 0;

(iii) one triplet imply V4 = U 2 = 0;
(iv) 3 couples imply V3 = 0;
(v) (v) 2 couples imply V5 = 0.

Remark 2.11. The above conditions depend only on the coefficients of the cubic
homogeneous parts of the system (2.1).

We rewrite the system (2.1) differently:

ẋ = a+ cx+ dy + gx2 + 2hxy + ky2 + px3 + 3qx2y + 3rxy2 + sy3

≡ P (x, y),

ẏ = b+ ex+ fy + lx2 + 2mxy + ny2 + tx3 + 3ux2y + 3vxy2 + wy3

≡ Q(x, y).

(2.4)

Let L(x, y) = Ux + V y + W = 0 be an invariant straight line of this family of
cubic system. Then, we have

UP (x, y) + V Q(x, y) = (Ux+ V y +W )(Ax2 + 2Bxy + Cy2 +Dx+ Ey + F ),

and this identity provides the following 10 relations:

Eq1 = (p−A)U + tV = 0, Eq6 = (2h− E)U + (2m−D)V − 2BW = 0,

Eq2 = (3q − 2B)U + (3u−A)V = 0, Eq7 = kU + (n− E)V − CW = 0,

Eq3 = (3r − C)U + (3v − 2B)V = 0, Eq8 = (c− F )U + eV −DW = 0

Eq4 = (s− C)U + V w = 0, Eq9 = dU + (f − F )V − EW = 0,

Eq5 = (g −D)U + lV −AW = 0, Eq10 = aU + bV − FW = 0.

(2.5)

It is well known that the infinite singularities (real or complex) of system (2.4)
are determined by the linear factors of the polynomial

C3 = yp3(x, y)− xq3(x, y).



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 17

Remark 2.12. Let C3 =
∏4
i=1(αix+ βiy), i = 1, 2, 3, 4. Since among the infinite

singularities we have the points at infinity of invariant lines, for an invariant line
Ux+ V y +W = 0 we must have [U : V ] = [αi : βi] for some i and we may assume
(U, V ) = (αi, βi). In this case, considering W as a fixed parameter, six equations
among (2.5) become linear with respect to the parameters {A,B,C,D,E, F} (with
the corresponding non-zero determinant) and we can determine their values, which
annihilate some of the equations (2.5)). So in what follows we will examine only
the non-trivial equations containing the last parameter W .

For the proof of the Main Theorem it is useful to consider the following homo-
geneous cubic system associated to system (2.4):

dx/dt = P3(x, y), dy/dt = Q3(x, y). (2.6)

Clearly in the case of four real distinct infinite singularities the polynomial
C3(x, y) has four distinct real linear factors. The following remark concerning the
associated homogeneous cubic system (2.6) is useful.

Remark 2.13. Assume that a cubic system (2.4) possesses invariant lines of total
multiplicity three (respectively two) in a real direction. Then the corresponding as-
sociated homogeneous cubic system (2.4) has one invariant line of total multiplicity
at least three (respectively two) in the same direction.

Indeed, if a system (2.4) possesses a triplet of parallel invariant lines (distinct or
coinciding) in a real direction then via an affine transformation this system could
be brought to the form

ẋ = x[(x+ b)2 + u], ẏ = Q(a, x, y).

It is clear that if u < 0 (respectively u > 0) then we have three real (respectively
one real and two complex) all distinct invariant lines. In the case u = 0 we either
have one simple and one double invariant lines if b 6= 0, or one triple invariant line
if b = 0. It remains to observe that in all four cases the corresponding associated
homogeneous cubic systems possess the invariant line x = 0 of total multiplicity at
least three. The case of a couple of parallel invariant lines ca be examined similarly.

According to [20, ] (see also [25]) we have the following result.

Lemma 2.14. If a cubic system (2.4) has 4 real distinct infinite singularities then
its associated homogeneous cubic system (2.6) could be brought via a linear trans-
formation to the canonical form (SI):

x′ = (p+ r)x3 + (s+ v)x2y + qxy2, C3 = xy(x− y)(rx+ sy),

y′ = px2y + (r + v)xy2 + (q + s)y3, rs(r + s) 6= 0
(2.7)

3. Proof of the main theorem

The proof proceeds in three steps:
Firstly we construct the cubic homogeneous parts (P̃3, Q̃3) of system (2.7) for

which the corresponding necessary conditions provided by Theorem 2.10 in order
to have the specified number of triplets or/and couples of invariant parallel lines in
the corresponding directions are satisfied.

Secondly we consider the system (2.4) with the cubic homogeneous parts (P̃3, Q̃3)
and generic homogeneities of lower degrees. For these system using the equations



18 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

(2.5) we determine the coefficients of these homogeneities in order to get the re-
quired number of invariant lines (of total multiplicity 7) in the required configu-
ration. Thus the second step ends with the construction of the canonical system
possessing the required configurations.

Thirdly, we prove that all the constructed configurations are distinct in view
of the Definition 1.3. To do this we apply some geometric invariants defined in
Subsection 3.6.

Assuming that cubic systems in the family (2.4) possess four distinct real infinite
singularities, according to Lemma 2.14 via a linear transformations they could be
brought to the family of systems

ẋ =a+ cx+ dy + gx2 + 2hxy + ky2 + (p+ r)x3 + (s+ v)x2y + qxy2,

ẏ =b+ ex+ fy + lx2 + 2mxy + ny2 + px2y + (r + v)xy2 + (q + s)y3
(3.1)

with C3 = xy(x − y)(rx + sy) and rs(r + s) 6= 0. As we have four real infinite
singularities and the total multiplicity of the invariant lines (including the line at
infinity) must be 7, then the above system could only have one of the following five
possible types of configurations of invariant lines:

(i) T = (3, 3); (ii) T = (3, 2, 1); (iii) T = (3, 1, 1, 1);

(iv) T = (2, 2, 2); (v) T = (2, 2, 1, 1).
(3.2)

3.1. Systems with the type of the configuration T = (3, 3). Since we have
two triplets of parallel invariant lines, according to Theorem 2.10 the conditions
V1 = V2 = U 1 = 0 are necessary for system (3.1). In [20, Section 5.1] it was
proved that in this case via a linear transformation and time rescaling the cubic
homogeneities of these system could be brought to the forms x3, y3 and we consider
the homogeneous system

ẋ = x3, ẏ = y3. (3.3)

So applying a translation we may assume g = n = 0 in the quadratic parts of
system (3.1) with the cubic homogeneities x3, y3. In such a way we obtain the
family of systems

ẋ = a+ cx+ dy + 2hxy + ky2 + x3,

ẏ = b+ ex+ fy + lx2 + 2mxy + y3,
(3.4)

for which we have C3(x, y) = xy(x− y)(x+ y).
To find out the directions of two triplets, according to Remark 2.13, we determine

the multiplicity of the invariant lines of system (3.3). For this system we calculate
(see the definition of the polynomial H(X,Y, Z) on the page 13, Notation 2.4):

H(X,Y, Z) = gcd(G1,G2,G3) = 3X3(X − Y )Y 3(X + Y ).

So system (3.3) has two triple invariant lines x = 0 and y = 0 and by Remark
2.13, system (3.4) could have triplets of parallel invariant lines only in these two
directions.

(i) The direction x = 0. Considering (2.5) and Remark 2.12 we obtain

Eq7 = k, Eq9 = d− 2hW, Eq10 = a− cW −W 3

and obviously we can have a triplet of parallel invariant lines (which could coincide)
in the direction x = 0 if and only if k = d = h = 0. Assuming that these conditions
hold we consider another direction for the second triplet.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 19

(ii)The direction y = 0. In this case we have

Eq5 = l, Eq8 = e− 2mW, Eq10 = b− fW −W 3

and again we conclude that for the existence of a triplet of parallel invariant lines
for system (3.4) the conditions e = l = m = 0 have to be satisfied.

It remains to examine the directions y = x and y = −x to determine the con-
ditions for the non-existence of an additional invariant line in one of these two
directions.

For the direction y = ±x considering (2.5) and Remark 2.12 we have

Eq7 = −3W ; Eq9 = ∓(c− f + 3W 2); Eq10 = a± b− cW −W 3.

We observe that in each one of the cases we could have only one invariant line.
Moreover the necessary and sufficient conditions for the existence of such a line are
c − f = a + b = 0 for the direction y = x and c − f = a − b = 0 for the direction
y = −x.

Thus we conclude that for the non-existence of an invariant line in additions to
the two triplets the following conditions are necessary and sufficient:

(c− f)2 + (a2 − b2)2 6= 0.

We arrive at the family of systems

ẋ = a+ cx+ x3, ẏ = b+ fy + y3 (3.5)

for which the above conditions must be satisfied. These system possess the invariant
lines defined by the equations

x3 + cx+ a = 0, y3 + fy + b = 0.

We observe that the number of distinct invariant lines and their kinds (real
and/or complex) depend on the discriminants of the cubic polynomials x3 + cx+ a
and y3 + fy + b, i.e.

ξ1 = −(27a2 + 4c3), ξ2 = −(27b2 + 4f3),

respectively. Moreover, we observe that the polynomial x3 + cx + a (respectively
y3 + fy + b) has a triple root if and only if ν1 = a2 + c2 = 0 (respectively ν2 =
b2 + f2 = 0).

Remark 3.1. Note that for system (3.5) we could not have simultaneously ν1 =
ν2 = 0, otherwise we obtain the homogeneous cubic system ẋ = x3, ẏ = y3 which
possesses invariant lines of total multiplicity nine.

In what follows we examine the possibilities provided by the discriminants ξ1
and ξ2.

(1) Case ξ1ξ2 > 0 and ξ1 + ξ2 > 0. Then each one of the mentioned cubic
polynomials factorizes in three distinct real factors, i.e. we obtain the system

ẋ = (x− α1)(x− β1)(x− δ1), ẏ = (y − α2)(y − β2)(y − δ2), (3.6)

where αi, βi, δi ∈ R, i = 1, 2. As all the lines are distinct then via the transformation

(x, y, t) 7→
(
α1 − (α1 − β1)x, α2 − (α1 − β1)y, t/(α1 − β1)2

)
(3.7)

we arrive at the following 3-parameter family of systems

ẋ = x(x− 1)(x− a), ẏ = y(y − b)(y − c), a(a+ 1)bc(b− c) 6= 0, (3.8)



20 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

where

a =
α1 − δ1
α1 − β1

, b =
α2 − β2
α1 − β1

, c =
α2 − δ2
α1 − β1

.

This system has 9 finite real singularities which are located at the intersections of
these two triplets of invariant lines. As a result we obtain Config. 7.1.

(2) Case ξ1ξ2 > 0 and ξ1 + ξ2 < 0. Then in each one of the directions x = 0
and y = 0, system (3.5) has one real and two complex invariant lines. After the
translation of the origin of coordinates at the intersections of the real invariant lines
we arrive at the system

ẋ = x(x2 + 2β1x+ δ1), ẏ = y(y2 + 2β2y + δ2),

where β2
1 − δ1 < 0 and β2

2 − δ2 < 0. So we can set β2
1 − δ1 = −u2 6= 0 and

β2
2 − δ2 = −v2 6= 0 respectively, the above system becomes

ẋ = x
[
u2 + (x+ β1)2

]
, ẏ = y

[
v2 + (y + β2)2

]
.

Since u 6= 0 then we may assume u = 1 due to the rescaling (x, y, t) 7→ (ux, uy, t/u2)
and we obtain the following 3-parameter family of systems

ẋ = x
[
(x+ a)2 + 1

]
, ẏ = y

[
(y + b)2 + c2

]
, c 6= 0. (3.9)

These systems possess 1 real and 8 complex finite singularities which are located
at the intersections of these two triplets of invariant lines (real and complex).

As a result, considering Notation 1.11 we obtain Config. 7.2 if ab 6= 0, Config.
7.3 if ab = 0 and a+ b 6= 0, and Config. 7.4 if a = 0 = b.

(3) Case ξ1ξ2 < 0. Without loss of generality we may assume ξ1 > 0 and ξ2 < 0
due to the change (x, y, t, a, b, c, f) 7→ (y, x, t, b, a, f, c) which conserves system (3.5).
Then we have the following factorization of the right hand parts of these system

ẋ = (x− α1)(x− β1)(x− δ1), ẏ = (y − α2)(y2 + 2β2y + δ2), (3.10)

where αi, βi, δi ∈ R, i = 1, 2 and β2
2 − δ2 < 0. So we can set β2

2 − δ2 = −u2 6= 0 and
then applying the transformation (3.7) we obtain the following 3-parameter family
of systems

ẋ = x(x− 1)(x− a), ẏ = y
[
(y + b)2 + c2

]
, a(a− 1)c 6= 0. (3.11)

These systems possess 3 real and 6 complex finite singularities which are located
at the intersections of these two triplets of invariant lines (real and complex).

So considering Notation 1.11 we obtain Config. 7.5 if b 6= 0, and Config. 7.6 if
b = 0.

(4) Case ξ1ξ2 = 0, ξ1 + ξ2 > 0, ν1ν2 6= 0. As it was mentioned earlier due
to the change (x, y, t, a, b, c, f) 7→ (y, x, t, b, a, f, c) (which conserves system (3.5))
we may assume ξ1 = 0 and ξ2 > 0. Following the same arguments as before,
system (3.5) can be written in the form (3.6) with β1 = α1 6= δ1. Then applying
the transformation (3.7) in which we substitute β1 by α1 6= δ1 we arrive at the
following 2-parameter family of systems

ẋ = x2(x− 1), ẏ = y(y − b)(y − c), bc(b− c) 6= 0. (3.12)

These systems possess three double real singularities (located at the intersections
of the double invariant line x = 0 with three simple lines) and 3 simple real sin-
gularities, located at the intersections of the simple invariant line x = 1 with the
triplet in the direction y = 0. As a result we obtain Config. 7.7.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 21

(5) Case ξ1ξ2 = 0, ξ1 + ξ2 < 0, ν1ν2 6= 0. We may assume again ξ1 = 0,
ξ2 < 0 and ν1 6= 0. In this case we consider system (3.10) with β1 = α1 6= δ1 and
following the same steps and the corresponding similar transformation we obtain
the following 2-parameter family of systems

ẋ = x2(x− 1), ẏ = y
[
(y + b)2 + c2

]
, c 6= 0. (3.13)

Clearly these systems possess three double singularities (one real and two complex)
on the double line x = 0 and three simple singularities (one real and two complex)
located on the simple invariant line x = 1.

So considering Notation 1.11 we obtain the configuration of invariant lines given
by Config. 7.8 if b 6= 0, and Config. 7.9 if b = 0.

(6) Case ξ1ξ2 = 0, ξ1 + ξ2 > 0, ν1ν2 = 0. As it was mentioned above we may
consider ξ1 = 0 which implies ξ2 > 0. Then ν2 6= 0 and hence we have ν1 = 0.
In this case we have a triple line in the direction x = 0 and after a translation we
obtain the system

ẋ = x3, ẏ = y(y − b)(y − c)
with bc 6= 0. Then applying the rescaling (x, y, t) 7→ (cx, cy, t/c2) we force c = 1
and we arrive at the following 1-parameter family of systems

ẋ = x3, ẏ = y(y − 1)(y − b), b(b− 1) 6= 0. (3.14)

It is easy to determine that these systems possess three triple real singularities
located on the triple invariant line x = 0. This leads to the configuration Config.
7.10.

(7) Case ξ1ξ2 = 0, ξ1 +ξ2 < 0, ν1ν2 = 0. So similarly as before, we may consider
ξ1 = ν1 = 0 and ξ2 < 0. In this case we have a triple line in the direction x = 0 and
after a translation setting some new parameters (see the second equation of system
(3.13)) we obtain the system

ẋ = x3, ẏ = y
[
c2 + (y + b)2

]
with c 6= 0. Then applying the rescaling (x, y, t) 7→ (cx, cy, t/c2) we arrive at the
following 1-parameter family of systems

ẋ = x3, ẏ = y
[
1 + (y + b)2

]
. (3.15)

These systems possess three triple singularities (one real and two complex) located
on the triple invariant line x = 0. Considering Notation 1.11 this leads to the
configuration Config. 7.11 if b 6= 0 and Config. 7.12 if b = 0.

(8) Case ξ1 = ξ2 = 0, ν1ν2 6= 0. Then we have two double real invariant lines
(one in the direction x = 0 and the second in the direction y = 0). Due to ν1ν2 6= 0
none of them could be triple. So after a translation which moves the origin of
coordinates at the intersection of the double lines we arrive at the system

ẋ = x2(x− a), ẏ = y2(y − b),
where ab 6= 0. Then applying the rescaling (x, y, t) 7→ (ax, ay, t/a2) we obtain the
following 1-parameter family of systems

ẋ = x2(x− 1), ẏ = y2(y − b), b 6= 0. (3.16)

It is not difficult to determine that these systems possess four distinct real finite
singularities: one of multiplicity four (located at the intersection of the double lines)
two double and one simple. As a result we obtain the configuration Config. 7.13.



22 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

(9) Case ξ1 = ξ2 = 0, ν1ν2 = 0. According to Remark 3.1 the condition ν21 +ν22 6=
0 is necessary and by the same reasons as above we may assume ν1 = 0 and ν2 6= 0.
Therefore we have a triple invariant line in the direction x = 0 and a double one in
the direction y = 0. As a result via a translation we obtain the system

ẋ = x3, ẏ = y2(y − a),

with a 6= 0. Then we may assume a = 1 due to the rescaling (x, y, t) 7→ (ax, ay, t/a2)
and we arrive at the system

ẋ = x3, ẏ = y2(y − 1). (3.17)

We observe that this system has only two distinct finite singularities: one of the
multiplicity six and one triple both located at the invariant line x = 0. So we obtain
Config. 7.14. Thus we have proved the following lemma.

Lemma 3.2. Assume that for a system (3.5) the conditions in terms of the poly-
nomials ξ1, ξ2, ν1 and ν2 and indicated below in the first column are satisfied. Then
this system could be brought via an affine transformation and time rescaling to one
of the corresponding canonical system, indicated in the second column. Moreover
this system possesses one of the configurations Config. 7.1 – 7.14 if and only if
the conditions on the parameters a and b of the corresponding canonical system
wherever they are indicated, are satisfied, respectively:

ξ1ξ2 > 0, ξ1 + ξ2 > 0 ⇒ (3.8) ⇔ Config. 7.1;

ξ1ξ2 > 0, ξ1 + ξ2 < 0 ⇒ (3.9) with


ab 6= 0 ⇔ Config. 7.2;

ab = 0, a+ b 6= 0 ⇔ Config. 7.3;

a = b = 0 ⇔ Config. 7.4;

ξ1ξ2 < 0 ⇒ (3.11) with

{
b 6= 0 ⇔ Config. 7.5;

b = 0 ⇔ Config. 7.6;

ξ1ξ2 = 0, ξ1 + ξ2 > 0, ν1ν2 6= 0 ⇒ (3.12) ⇔ Config. 7.7;

ξ1ξ2 = 0, ξ1 + ξ2 < 0, ν1ν2 6= 0 ⇒; (3.13) with

{
b 6= 0 ⇔ Config. 7.8;

b = 0 ⇔ Config. 7.9;

ξ1ξ2 = 0, ξ1 + ξ2 > 0, ν1ν2 = 0 ⇒ (3.14) ⇔ Config. 7.10;

ξ1ξ2 = 0, ξ1 + ξ2 < 0, ν1ν2 = 0 ⇒ (3.15) with

{
b 6= 0 ⇔ Config. 7.11;

b = 0 ⇔ Config. 7.12;

ξ1ξ2 = 0, ξ1 + ξ2 = 0, ν1ν2 6= 0 ⇒ (3.16) ⇔ Config. 7.13;

ξ1ξ2 = 0, ξ1 + ξ2 = 0, ν1ν2 = 0 ⇒ (3.17). ⇔ Config. 7.14.

3.2. Systems with configuration type T = (3, 2, 1). In this subsection we con-
struct the cubic system with 4 real infinite singular points which has 6 invariant
affine straight lines (counted with multiplicities) with configuration of type (3, 2, 1),
having total multiplicity 7 including the line at infinity.

Since we have one triplet and one couple of parallel invariant lines, according to
Theorem 2.10 the conditions V4 = V5 = U2 = 0 are necessary for system (3.1). In
[6, Subsection 3.3.1] it was proved that in this case via a linear transformation and
time rescaling the associated cubic homogeneous system could be brought to the



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 23

form

ẋ = rx3, ẏ = (r − 1)xy2 + y3, r(r + 1) 6= 0. (3.18)

Consider the generic cubic system with cubic homogeneities as indicated in sys-
tem (3.18). Since r 6= 0 via a translation we may assume g = n = 0 in system (3.1),
i.e. a system possessing invariant lines in the configuration (3, 2, 1) can be brought
to the following family:

ẋ = a+ cx+ dy + 2hxy + ky2 + rx3, r(r + 1) 6= 0,

ẏ = b+ ex+ fy + lx2 + 2mxy + (r − 1)xy2 + y3.
(3.19)

Remark 3.3. We observe that due to a rescaling, the directions y = −rx and
y = x can be interchanged without changing the form of the system (3.19).

First of all we recall that the directions for the potential invariant lines of system
(3.44) are defined by the factors of the invariant polynomial C3(x, y) = xy(x −
y)(rx+y). Since all non-zero coefficients of the polynomials of degree less than three
of the right-hand parts of system (3.19) are free parameters, we can assume that any
rescaling does not affect them. So we consider only the cubic homogeneous system
associated to the system (3.19). Due to r 6= 0 applying the rescaling (x, y, t) 7→
(−x1, ry1, t1/r2) we obtain the homogeneous cubic system

ẋ1 = r1x
3
1, ẏ1 = (r1 − 1)x1y

2
1 + y31 ,

where r1 = 1/r. Since x1 = −x and y1 = y/r for these system we have

C3(x1, y1) = x1y1(x1 − y1)(r1x1 + y1)

= −x(y/r)(−x− y/r)(−x/r + y/r)

= − 1

r3
xy(rx+ y)(x− y).

So we conclude that the direction rx+y = 0 (respectively x−y = 0) corresponding
to system (3.19) passes to the direction x1 − y1 = 0 (respectively rx1 + y1 = 0)
corresponding to the above system and hence the claim of Remark 3.3 is valid.

In what follows we determine necessary and sufficient conditions for a system
(3.19) to have configuration of the type T = (3, 2, 1).

According to [6, Remark 3.3] for system (3.19) the following remark is valid:

Remark 3.4. A cubic system (3.19) can possess: (i) a triplet of parallel invariant
lines either in the direction x = 0, or y = 0 and in the second case the condition
r = 1 holds; (ii) a couple of parallel invariant lines either in the direction y = 0, or
y = x (if r = −1/2), or y = 2x (if r = −2).

So we have to examine the two cases: (r− 1)(1 + 2r)(2 + r) 6= 0 and (r− 1)(1 +
2r)(2 + r) = 0 (which splits in tree subcases).

3.2.1. Case (r − 1)(1 + 2r)(2 + r) 6= 0. Then by Remark 3.4 system (3.19) could
have a triplet (respectively a couple) of parallel invariant lines only in the direction
x = 0 (respectively y = 0). In [6] (see Subsection 3.2.1.1) these two directions were
examined and the following result was obtained:

Lemma 3.5. In the case r− 1 6= 0 system (3.19) has a triplet of parallel invariant
lines (which could be real or complex, distinct or coinciding) in the direction x =



24 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

0 and a couple of such lines in the direction y = 0 if and only if the following
conditions hold:

k = d = h = l = 0, e =
[
4m2 + f(r − 1)2

]
/(r − 1),

b = −2m
[
4m2 + f(r − 1)2

]
/(r − 1)3.

(3.20)

Clearly, in addition we need exactly one invariant line, which could be in one of
the directions y = x or rx+y = 0. However according to Remark 3.3 it is sufficient
to only consider the direction y = x.

Considering the conditions (3.20) in [6, Subsection 3.3.2] was proved that a
system (3.19) possesses one invariant line in this direction if and only if beside the
conditions (3.20) the following conditions are satisfied:

c = fr +
12m2r(2 + r)

(r − 1)(1 + 2r)2
, a = − 6fmr

(r − 1)(1 + 2r)
− 72m3r(1 + r + r2)

(r − 1)3(1 + 2r)3
. (3.21)

It can be checked directly that system (3.19) with the conditions (3.20) and
(3.21), possess the following invariant lines with configuration of the type T =
(3, 2, 1):

L1 = (r − 1)(2r + 1)x− 6m, L2 = (1 + 2r)(x− y)− 2m,

L3,4 = (r − 1)2(2r + 1)2x2 + 6m(r − 1)(2r + 1)x

+ f(1 + r − 2r2)2 + 12m2(1 + r + r2),

L5,6 = (r − 1)2y2 + 2m(r − 1)y + f(r − 1)2 + 4m2.

To determine if the invariant lines L3,4 = 0 and L5,6 = 0 are real or complex as well
as if the invariant line L1 = 0 coincides with one of the lines L3,4 = 0 we calculate:

Discrim[L3,4, x] = −4(r − 1)2(1 + 2r)4
[
3m2 + f(r − 1)2

]
,

Discrim[L5,6, y] = −4(r − 1)2
[
3m2 + f(r − 1)2

]
,

Resx(L1, L3,4) = (r − 1)2(1 + 2r)2
[
f(r − 1)2(1 + 2r)2 + 12m2(7 + r + r2)

]
,

Setting the notation

ξ = −
[
3m2 + f(r − 1)2

]
, ν = f(r − 1)2(1 + 2r)2 + 12m2(7 + r + r2) (3.22)

we observe that

sign
(

Discrim[L3,4, x]
)

= sign
(

Discrim[L5,6, y]
)

= sign(ξ).

Moreover, if ξ = 0 then ν 6= 0, otherwise the condition ξ = 0 = ν imply m =
f = 0 and we arrive at the homogeneous system possessing invariant lines of total
multiplicity 8. So we examine the possibilities provided by the polynomials ξ and
ν.

(a) Case ξ > 0 and ν 6= 0. This means that 3m2 + f(r − 1)2 < 0, i.e. all 6

invariant lines are real. Setting 3m2 +f(r−1)2 = −u2 6= 0 we obtain f = − 3m2+u2

(r−1)2
and we arrive at the following system

ẋ =r
[
x− 6m

(r − 1)(1 + 2r)

][
x+

3m− u− 2ru

(r − 1)(1 + 2r)

][
x+

3m+ u+ 2ru

(r − 1)(1 + 2r)

]
,

ẏ =
[
y +

m− u
(r − 1)

][
y +

m+ u

(r − 1)

][
(r − 1)x+ y − 2m

(r − 1)

] (3.23)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 25

For f = − 3m2+u2

(r−1)2 we obtain

ν = (9m− u− 2ru)(9m+ u+ 2ru) 6= 0.

Then considering the condition u(r − 1)(1 + 2r)(2 + r) 6= 0 via the transformation

(x, y, t) 7→
(6m+ (u− 9m+ 2ru)x

(r − 1)(1 + 2r)
,

(u−m)(1 + 2r) + (u− 9m+ 2ru)y

(r − 1)(1 + 2r)
,

(r − 1)2(1 + 2r)2

(u− 9m+ 2ru)2
t
)
,

after setting a new parameter

a =
9m+ u+ 2ru

9m− u− 2ru

(
i.e. m =

(1 + a)(1 + 2r)u

9(a− 1)

)
,

system (3.23) can be brought to the 2-parameter family of systems:

ẋ = rx(x− 1)(x− a), a(a− 1)r(r − 1) 6= 0

ẏ = y(y + 1− a)
[
(r − 1)x+ y + 1

]
.

(3.24)

These systems possess the invariant lines

L1 : x = 0, L2 : x = 1, L3 : x = a,

L4 : y = 0, L5 : y = a− 1, L6 : x− y = 1

and the nine finite singularities:

M1(0, 0), M2(0,−1), M3(0, a− 1), M4(1, 0), M5(1,−r),
M6(1, a− 1), M7(a, 0), M8(a, a− 1), M9

(
a, a(1− r)− 1

)
.

It is easy to determine that 7 of these singularities are located at the intersec-
tions of the above invariant lines, more precisely the singular points Mi for i ∈
{1, 2, 3, 4, 6, 7, 8}. The singular point M5 (respectively M9) is located on the invari-
ant line L2 (respectively L3). Moreover we have two singular points located at the
intersections of three invariant lines: L2, L4 and L6 which intersect at the point
M4, whereas L3, L5 and L6 intersect at the point M8. To determine all possible
configurations for system (3.24) we have to examine the positions of the invariant
lines as well as of the singularities M5 and M9 depending on the parameters a and r.

Let us first examine the position of the invariant lines. We observe that four
of the lines are fixed and only the positions of the lines L3 and L5 depend on the
parameter a. More exactly, if a < 0 then the non-fixed invariant line x = a is
located on the left of the fixed invariant lines x = 0 and x = 1. If 0 < a < 1 then
the invariant line x = a is located between the invariant lines x = 0 and x = 1.
And finally, if a > 1 then the invariant line x = a is located on the right of the
invariant lines x = 0 and x = 1.

Notation 3.6. Assume that the two finite real singular points M̃1(x1, y1) and

M̃2(x2, y2) of a cubic system are located on the real invariant line ax+ by + c = 0

of this system. Then: (α) in the case a 6= 0 we say that the singular point M̃1 is

located below (respectively above) or coincides with the singularity M̃2 if y1 ≤ y2
(respectively y2 < y1) and we denote this position by M̃1 � M̃2 (respectively

M̃2 ≺ M̃1);

(β) in the case a = 0 (then y1 = y2) we say that the singular point M̃1 is located

on the left or coincides with (respectively on the right) the singularity M̃2 if x1 ≤ x2



26 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

(respectively x2 < x1) and we again denote this position by M̃1 � M̃2 (respectively

M̃2 ≺ M̃1).

Next we consider the position of the finite singularity M5(1, y5) with y5 = −r
(respectivelyM9(a, y9) with y9 = a(1−r)−1 on the invariant line x = 1 (respectively
x = a). We observe that on the line x = 1 (respectively x = a) two real singularities
M4(1, 0) and M6(1, a−1) (respectively M7(a, 0)) and M8(a, a−1)) are located and
their reciprocal position depends on the sign of a − 1 6= 0. So we distinguish two
possibilities: (i) a < 1 and (ii) a > 1.

It is clear that the positions of these singularities in two different cases (i) and
(ii) could be distinct and therefore we examine each one of these cases. Since ar 6= 0
we deduce that y5 6= 0 (respectively y9 6= a − 1), but we could have y5 = a − 1
(respectively y9 = 0) and in this case the singularity M5 (respectively M9) coalesced
with the singularity M6 (respectively M7).

Case (i): a < 1. Then we have M6 ≺ M4 and M8 ≺ M7 and considering the
coordinates y5 = −r and y9 = a(1− r)− 1 and Notation 3.6 we have the following
implications.
(I) For the singular point M5:

y5 ≤ a− 1 ⇒ M5 �M6 ≺M4; a− 1 < y5 < 0 ⇒ M6 ≺M5 ≺M4;

y5 > 0 ⇒ M6 ≺M4 ≺M5.

(II) For the singular point M9:

y9 < a− 1 ⇒ M9 ≺M8 ≺M7; a− 1 < y9 < 0 ⇒ M8 ≺M9 ≺M7;

y9 ≥ 0 ⇒ M8 ≺M7 �M9.

Case (ii): a > 1. Then we have M4 ≺ M6 and M7 ≺ M8, and considering the

coordinates y5 and y9 we have the following implications.
(I) For the singular point M5:

y5 < 0 ⇒ M5 ≺M4 ≺M6; 0 < y5 < a− 1 ⇒ M4 ≺M5 ≺M6;

y5 ≥ a− 1 ⇒ M4 ≺M6 �M5.

(II) For the singular point M9:

y9 ≤ 0 ⇒ M9 �M7 ≺M8; 0 < y9 < a− 1 ⇒ M7 ≺M9 ≺M8;

y9 > a− 1 ⇒ M7 ≺M8 ≺M9.

It is clear that not all the possibilities described above are realizable and exam-
ining the compatibilities of the conditions it is not too hard to convince ourselves
(using, for example, the tools ”‘FindInstance” or ”‘Reduce” of computer algebra
system Mathematica) that the following lemma is valid.

Lemma 3.7. Consider the family of systems (3.19) with the conditions (3.20),
(3.21), (r − 1)(1 + 2r)(2 + r) 6= 0 and ξ > 0 and ν 6= 0. Then via an affine
transformation and time rescaling, after introducing some new parameters, system
(3.19) can be brought to the 2-parameter family of systems (3.24). Moreover this
family of systems possesses the following configurations of invariant lines when the
corresponding conditions indicated below, are satisfied (examples are given in the



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 27

last column):

Config. 7.15 ⇔;

{
a < 0, 1− a < r < 1− 1/a, (a = −1/2, r = 2),

or a < 0, 1− 1/a < r < 1− a (a = −2, r = 2);

Config. 7.16 ⇔

{
−1 < a < 0, r = 1− 1/a (a = −1/2, r = 3),

or a < −1, r = 1− a (a = −2, r = 3);

Config. 7.17 ⇔ a < 0, r > 1− a, r > 1− 1/a (a = −2, r = 4);

Config. 7.18 ⇔

{
−1 < a < 0, r = 1− a (a = −1/4, r = 5/4),

or a < −1, r = 1− 1/a (a = −2, r = 3/2),

Config. 7.19 ⇔ a = −1, r = 2;

Config. 7.20 ⇔ a < 0, 0 < r < 1− a, r < 1− 1/a; (a = −2, r = 1/2);

Config. 7.21 ⇔ a < 0, r < 0; (a = −1, r = −3);

Config. 7.22 ⇔

{
0 < a < 1, r > 1− a, (a = 1/2, r = 3/4),

or a > 1, r > 1− 1/a (a = 5/4, r = 1/2);

Config. 7.23 ⇔

{
0 < a < 1, r = 1− a (a = 1/2, r = 1/2),

or a > 1, r = 1− 1/a, (a = 2, r = 1/2);

Config. 7.24 ⇔′
{

0 < a < 1, 0 < r < 1− a (a = 1/2, r = 1/4),

or a > 1, 0 < r < 1− 1/a (a = 3, r = 1/2);

Config. 7.25 ⇔

{
0 < a < 1, r = 1− 1/a (a = 5/32, r = −27/5),

or a > 1, r = 1− a (a = 4, r = −3);

Config. 7.26 ⇔

{
0 < a < 1, r < 1− 1/a (a = −1/4, r = −4),

or a > 1, r < 1− a; (a = 3, r = −3);

Config. 7.27 ⇔

{
0 < a < 1, 1− 1/a < r < 0, (a = 1/4, r = −5/2),

or a > 1, 1− a < r < 0(a = 5, r = −3).

(b) Case ξ > 0 and ν = 0. Since (r − 1)(2r + 1) 6= 0, considering (3.22) the

condition ν = 0 gives f = − 12m2(7+r+r2)
(r−1)2(1+2r)2 and we arrive at the system

ẋ =r
[
x− 6m

(r − 1)(1 + 2r)

]2[
x+

12m

(r − 1)(1 + 2r)

]
,

ẏ =
[
y +

2m(−4 + r)

(r − 1)(1 + 2r)

][
y +

2m(5 + r)

(r − 1)(1 + 2r)

][
(r − 1)x+ y − 2m

(r − 1)

]
We observe that m 6= 0 otherwise we obtain f = 0 and this implies ξ = 0, i.e. we
arrive at a contradiction. Then considering the condition m(r−1)(1+2r)(2+r) 6= 0
via the transformation

(x, y, t) 7→
( 6m(1− 3x)

(r − 1)(1 + 2r)
,

2m(4− r − 3y)

(r − 1)(1 + 2r)
,

(r − 1)2(1 + 2r)2

324m2
t
)

we arrive at the following 1-parameter family of systems

ẋ = rx2(x− 1), ẏ = y(y − 1)
[
(r − 1)x+ y

]
(3.25)



28 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

These system possess the invariant lines

L1,2 : x = 0 (double), L3 : x = 1, L4 : y = 0, L5 : y = 1, L6 : y = x

and the following five finite singularities of total multiplicity 9:

M1,2,3,4(0, 0), M5,6(0, 1), M7(1, 0), M8(1, 1), M9(1, 1− r).
As we can see the positions of the invariant lines are fixed, as well as the positions
of the finite singularities, except for the singularity M9(1, 1 − r) located on the
invariant line x = 1. It is not too difficult to detect that depending on the position
of M9 with respect to the singularities M7 and M8 as well as of the position of the
s-points (”smooth”, see Definition 3.29 on page 103) infinite singularity we obtain
the following four distinct singularities: Config. 7.28 if r < −1; Config. 7.29 if
−1 < r < 0; Config. 7.30 if 0 < r < 1; and Config. 7.31 if r > 1.

(c) Case ξ < 0. This means that 3m2+f(r−1)2 > 0, i.e. the invariant lines L3,4

and L5,6 are complex, whereas L1 and L2 remain real. Setting 3m2 + f(r − 1)2 =

u2 6= 0 we obtain f = − 3m2−u2

(r−1)2 and we arrive at the following system

ẋ =r
[
x− 6m

(r − 1)(1 + 2r)

][(
x+

3m

(r − 1)(1 + 2r)

)2
+

u2

(r − 1)2

]
,

ẏ =
[(
y +

m

(r − 1)

)2
+

u2

(r − 1)2

][
(r − 1)x+ y − 2m

(r − 1)

]
For f = − 3m2−u2

(r−1)2 we obtain ν = 81m2 + u2(1 + 2r)2 6= 0 because of u(1 + 2r) 6= 0.

So considering the condition u(r − 1)(1 + 2r)(2 + r) 6= 0 via the transformation

(x, y, t) 7→
(
− u(2a− 3x)

3(r − 1)(1 + 2r)
,
u(a+ 2ar + 9x)

9(r − 1)
,

(r − 1)2

u2
t
)

and setting a new parameter a = − 9m
u(1+2r) (i.e. m = −a(1 + 2r)u/9) we arrive at

the following 2-parameter family of systems:

ẋ = rx
[
(x− a)2 + 1

]
, ẏ = (1 + y2)

[
(r − 1)x+ y + a

]
. (3.26)

These systems possess the following six invariant affine lines (two real and four
complex):

L1 : x = 0, L2,3 : x = a± i, L4,5 : y = ±i, L6 : x− y = a

and the following nine finite singularities:

M1(0,−a), M2,3(0,±i), M4,5(a± i, i), M6,7(a± i,−i),
M8(a− i,−ar + i(r − 1)), M9(a+ i,−ar − i(r − 1)).

We observe that one of the infinite singularities is located at the end of the affine
line y = −rx and since the other three fixed infinite singularities ale located at the
intersections of the line at infinity with the lines x = 0, y = 0 and y = x it is clear
that we need to distinguish three possibilities: r < −1, −1 < r < 0 and r > 0,
respectively. As a result, considering Notation 1.11, in the case a 6= 0 we obtain
Config. 7.32 if r < −1; Config. 7.33 if −1 < r < 0; and Config. 7.34 if r > 0.

On the other hand if a = 0 we obtain Config. 7.35 if r < −1; Config. 7.36 if
−1 < r < 0 and Config. 7.37 if r > 0.

(d) Case ξ = 0. Considering (3.22) we obtain f = −3m2/(r − 1)2 and then
ν = 81m2. We observe that m 6= 0 otherwise we obtain the system (3.18) possessing



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 29

invariant lines of total multiplicity 8, because for these system we have (see Notation
2.4):

H(X,Y, Z) = gcd(G1,G2,G3) = −rX3(X − Y )Y 2(rX + Y ).

So for f = −3m2/(r − 1)2 we arrive at the family of systems

ẋ =r
[
x+

3m

(r − 1)(1 + 2r)

]2[
x− 6m

(r − 1)(1 + 2r)

]
,

ẏ =
[
y +

m

r − 1

]2[
(r − 1)x+ y − 2m

(r − 1)

]
.

Since m 6= 0 we can apply the transformation

(x, y, t) 7→
( 3m(3x− 1)

(r − 1)(1 + 2r)
, −m(1 + 2r − 9y)

(r − 1)(1 + 2r)
,

(r − 1)2(1 + 2r)2

81m2
t
)

we arrive at the 1-parameter family of systems

ẋ = rx2(x− 1), ẏ = y2
[
(r − 1)x+ y − r

]
.

These systems possess the invariant lines

L1,2 : x = 0 (double), L3 : x = 1, L4,5 : y = 0, (double), L6 : y = x

and the following four finite singularities of total multiplicity 9:

M1,2,3,4(0, 0), M5,6(0, r), M7,8(1, 0), M9(1, 1).

As we can see, the positions of the invariant lines are fixed, as well as the positions
of the finite singularities, except for the double singularity M5,6(0, r) located on the
invariant line x = 0. It is not too difficult to detect that depending on the position
of M5,6 with respect to the singularity M1 as well as the position of the s-points at
infinity, due to r(r + 1) 6= 0, we obtain the following three distinct configurations:
Config. 7.38 if r < −1; Config. 7.39 if −1 < r < 0 and Config. 7.40 if r > 0.

3.2.2. Case (r− 1)(1 + 2r)(2 + r) = 0. We examine each one of the three subcases
given by these factors. However we observe that the subcase r = −2 could be
brought to the subcase r = −1/2 via the rescaling x→ x/2 . Therefore we consider
only two subcases: r = 1 and r = −1/2.

For system (3.19), considering the equations (2.5) and Remark 2.12 we obtain

Eq7 = k, Eq9 = d− 2hW, Eq10 = a− cW − rW 3, (3.27)

for the direction x = 0,

Eq′5 = l, Eq′8 = e− 2mW + (r − 1)W 2, Eq′10 = b− fW −W 3 (3.28)

for the direction y = 0 and

Eq′′6 = l − 2h− k + 2m− (1 + 2r)W,

Eq′′9 = −c− d+ e+ f + 2(l − h+m)W − 3rW 2,

Eq′′10 = −a+ b+ (e− c)W + lW 2 − rW 3

(3.29)

for the direction y = x.

1. Subcase r = 1. According to Remark 3.4 system (3.19) could have the
following parallel invariant lines: (i) a triplet in the direction x = 0 and a couple
in the direction y = 0; (ii) a triplet in the direction y = 0 and a couple in the
direction x = 0.



30 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

However for r = 1 from (3.27) and (3.28) it follows that in each one of these
directions we could have either one, or three parallel invariant lines. So in the case
r = 1 we could not have a configuration with the type T = (3, 2, 1).

2. Subcase r = −1/2. According to Remark 3.4 system (3.19) could have a
triplet of parallel invariant lines in the direction x = 0 and a couple of parallel
invariant lines either in the direction y = 0 or y = x.

Forcing to have a triplet in the direction x = 0, according to (3.27), for system
(3.19) the conditions k = d = h = 0 must be satisfied. So it remains to examine
two possibilities for the existence of a couple of invariant lines: (i) in the direction
y = 0 and (ii) in the direction y = x.

2.1. Possibility: a couple in the direction y = 0. Considering (3.28) we have
l = 0 and taking into account the condition r = −1/2 we calculate

R
(1)
W (Eq′8, Eq

′
10) = −(6e+ 9f + 16m2)/4 = 0 ⇒ e = −(9f + 16m2)/6

and then we obtain

R
(0)
W (Eq′8, Eq

′
10) = −(27b− 36fm− 64m3)2/216 = 0, ⇒ b = 4m(9f + 16m2)/27.

So we determine the conditions

r = −1/2, k = d = h = l = 0, e = −(9f +16m2)/6, b = 4m(9f +16m2)/27 (3.30)

which lead to the family of systems

ẋ = (2a+ 2cx− x3)/2,

ẏ = (8m− 9x+ 6y)(9f + 16m2 − 12my + 9y2)/54.
(3.31)

We observe that these systems possess one triplet in the direction x = 0 and one
couple of parallel invariant lines in the direction y = 0. So we need one more
invariant affine line, which could be either in the direction y = x or in the direction
y = x/2 (since we have r = −1/2). However according to Remark 3.3 it is sufficient
to only examine the case y = x.

Considering (3.29) and the conditions (3.30) we obtain Eq′′6 = 2m = 0, i.e.
m = 0 and then we calculate

R
(0)
W (Eq′′9 , Eq

′′
10) =

[
27a2 − 4(2c+ f)(c+ 2f)2

]
/8 ≡ Ψ′′(a, c, f).

In order for the equation Ψ′′ = 0 to have real solutions we clearly must have either
2c+ f ≥ 0 or c+ 2f = 0. We consider each one of these cases.

2.1.1. Case 2c + f ≥ 0. So setting 2c + f = 3u2 ≥ 0 we obtain f = −2c + 3u2

and then we have

Ψ′′ = 27(a+ 2cu− 4u3)(a− 2cu+ 4u3) = 0.

So because of the change u→ −u we may assume a− 2cu+ 4u3 = 0 and we obtain
the condition a = 2u(c − 2u2). Considering the conditions (3.30) we arrive at the
family of systems

ẋ = (2u+ x)(2c− 4u2 + 2ux− x2)/2, ẏ = (3x− 2y)(2c− 3u2 − y2)/2. (3.32)

These systems possess the following invariant lines with the configuration of the
type T = (3, 2, 1):

L1 = x+ 2u, L2 = x− y − u,
L3,4 = (u− x)2 − (2c− 3u2), L5,6 = y2 − (2c− 3u2).

(3.33)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 31

Clearly the invariant lines L3,4 = 0 as well as L5,6 = 0 are real (respectively
coinciding) if 2c − 3u2 > 0 (respectively 2c − 3u2 = 0) and they are complex if
2c− 3u2 < 0. Moreover since

Resx(L1, L3,4) = −2(c− 6u2),

we conclude, that the invariant line L1 = 0 coincides with one of the lines L3,4 = 0
if and only if c = 6u2 and clearly this implies 2c − 3u2 = 3u2 > 0, i.e. the
invariant lines L3,4 = 0 must be real. So we examine the subcases provided by the
polynomials ξ′′ = 2c− 3u2 and ν′′ = c− 6u2.

2.1.1.1. Subcase ξ′′ > 0 and ν′′ 6= 0. Then all the invariant lines (3.33) are real
and distinct. Setting 2c− 3u2 = v2 6= 0 we obtain c = (3u2 + v2)/2 and we arrive
at the system

ẋ = (2u− x)(u− v + x)(u+ v + x)/2, ẏ = (3x− 2y)(v − y)(v + y)/2. (3.34)

Then since v 6= 0 via the transformation

(x, y, t) 7→
(
− u+ v − 2vx, −v(1 + 2y), t/(2v2

)
and introducing a new parameter b = (v − 3u)/(2v) (i.e. u = v(1− 4a)/3) system
(3.34) can be brought to the system

ẋ = x(x− 1)(b− x), ẏ = y(y + 1)(2 + b− 3x+ 2y).

We observe that in system (3.24) the condition a − 1 6= 0 holds. Then setting
b = a/(a− 1) via the following affine transformation and time rescaling:

x1 = (1− a)x+ a, y1 = (1− a)y, t1 =
2

(1− a)2
t

the above system could be brought to the system (3.24) for which r = −1/2. This
family was investigated earlier (see Lemma 3.7) and since for r = −1/2 we do
not have bifurcation points for the family (3.24) we deduce that there are no new
configurations.

2.1.1.2. Subcase ξ′′ > 0 and ν′′ = 0. Then c = 6u2 which implies ξ′′ = 6u2 > 0
and we obtain the family of systems

ẋ = −(2u− x)2(4u+ x)/2, ẏ = (3x− 2y)(3u− y)(3u+ y)/2.

Due to u 6= 0, the above system can be brought via the transformation

(x, y, t) 7→
(
2u(1− 3x), 3u(2y − 1), t/(18u2)

)
to the system

ẋ = −x2(x− 1), ẏ = y(y − 1)(3x+ 2y − 2).

After the additional change (x, y, t) 7→ (x, 1−y, t/2) this system can be transformed
to the system (3.25) with r = −1/2. This system was already investigated and hence
no new configuration could be obtained.

2.1.1.3. Subcase ξ′′ < 0. Setting 2c−3u2 = −v2 < 0 we obtain c = (3u2 +v2)/2
and we arrive at the following system

ẋ = (2u− x)
[
(x+ u)2 + v2

]
/2, ẏ = (3x− 2y)(y2 + v2)/2. (3.35)

Since v 6= 0 the above system could be brought via the transformation (x, y, t) 7→(
2u+ vx, vy, −2t/v2

)
to the system

ẋ = x
[
(x− a)2 + 1

]
, ẏ = (1 + y2)(3x− 2y − 2a).



32 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

We observe that applying the time rescaling t→ −t/2, the above system becomes
the system (3.26) with r = −1/2 and hence, no new configuration could be obtained.

2.1.2. Case c+ 2f = 0. Then the equation Ψ′′ = 0 implies a = 0 and we arrive
at the family of systems

ẋ = −x(4f + x2)/2, ẏ = −(3x− 2y)(f + y2)/2. (3.36)

These systems possess the following 8 invariant affine lines

x = 0, 4f + x2 = 0, x = 2y, f + y2 = 0, f + (x− y)2 = 0,

i.e. we are out of the class of system studied in this article.

2.2. Possibility: a couple in the direction y = x. Considering (3.29) and the
conditions r = −1/2 and k = d = h = 0 (in order to have a triplet in the direction
x = 0) we obtain:

Eq′′7 = l + 2m, Eq′′9 = −c− d+ e+ f + 2lW − 3W 2/2,

Eq′′10 = −a+ b+ (e− c)W + lW 2 +W 3/2.

So the condition Eq′′7 = 0 yields l = −2m which implies

R
(1)
W (Eq′′9 , Eq

′′
10) = (−6c+ 6e− 3f − 16m2)/4 = 0 ⇒ e = (6c+ 3f + 16m2)/6

and then we obtain

R
(0)
W (Eq′′9 , Eq

′′
10) = (27a− 27b− 36fm− 64m3)2/216 = 0,

⇒ b = (27a− 36fm− 64m3)/27.

So we found that under the conditions

r = −1/2, k = d = h = 0, l = −2m,

e = (6c+ 3f + 16m2)/6, b = (27a− 36fm− 64m3)/27,
(3.37)

system (3.19) has one triplet in the direction x = 0 and one couple of parallel
invariant lines in the direction y = x. In this case we arrive at the system

ẋ = (2a+ 2cx− x3)/2,

ẏ =
1

27
(27a− 36fm− 64m3) +

1

6
(6c+ 3f + 16m2)x− 2mx2 + fy

+ 2mxy − 3

2
xy2 + y3.

However we detect that these system can be brought via the change (x, y, t) 7→
(x, x − y, t) to the system (3.31) with exactly the same parameters a, c, f and m.
Since the system (3.31) were already investigated we deduce that no new configu-
rations could be obtained.

3.3. Systems with the configuration of the type T = (3, 1, 1, 1). In this sub-
section we construct a cubic system with 4 real infinite singular points which has
6 invariant affine straight lines, with configuration of type T = (3, 1, 1, 1), having
total multiplicity 7, as always the invariant straight line at infinity included.

According to Theorem 2.10 if a cubic system possesses 6 invariant affine straight
lines in the configuration of type T = (3, 1, 1, 1), then necessarily condition V4 =
U2 = 0 holds.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 33

3.3.1. Construction of the associated homogeneous cubic system. As a first step
we construct the cubic homogeneous parts of system (3.1) for which the above
condition is fulfilled. Since we have 4 real infinite distinct singularities, according
to Lemma 2.14 we consider the family of systems

ẋ =(p+ r)x3 + (s+ v)x2y + qxy2,

ẏ =px2y + (r + v)xy2 + (q + s)y3, rs(r + s) 6= 0,
(3.38)

and we force the condition V4 = U2 = 0 to be satisfied.
A straightforward computation of the value of V4 for system (3.38) yields V4 =

−9216 V̂4 C3(x, y), where

V̂4 = r2(3q + s+ v) + r(2pq − s2 + 3qv + v2)− s(2pq + 3ps+ 3pv + sv + v2).

As for system (3.38) we have C3 = xy(x − y)(rx + sy) 6= 0, we conclude that for

these systems the condition V4 = 0 is equivalent to V̂4 = 0.
We observe that the invariant polynomial U2 is a homogeneous polynomial of

degree four in x and y. So we shall use the following notations:

U2 =

4∑
j=0

U2jx4−jyj .

Calculating the value of the polynomial U2 for system (3.38) we obtain

U20 = p
[
q(2p+ 3r)2 + (s+ v)(2pr + 3r2 − 3ps− 3rs− pv)

]
≡ pÛ20

and we consider two cases: p 6= 0 and p = 0.

1. Case p 6= 0. Then we must have Û20 = 0 and as this polynomial is linear with
respect to the parameter q, we examine two subcases: 2p+ 3r 6= 0 and 2p+ 3r = 0.

1.1. Subcase 2p+ 3r 6= 0. In this case the condition Û20 = 0 gives

q = − (s+ v)(2pr + 3r2 − 3ps− 3rs− pv)

(2p+ 3r)2

and then we calculate

U21 =
3p(s+ v)(2p+ 3r + s+ v)(2ps+ 2rs− rv)

2p+ 3r
= −(2p+ 3r)V̂4.

So because of 2p+ 3r 6= 0 the condition U21 = 0 implies V̂4 = 0.
On the other hand since p 6= 0 the condition U20 = 0 gives

(s+ v)(2p+ 3r + s+ v)(2ps+ 2rs− rv) = 0

and we consider the three possibilities provided by this condition.

1.1.1. Possibility s + v = 0. Then v = −s (this implies q = 0) and we obtain

U2 = 0 = V̂4. In this case we arrive at the system

ẋ = (p+ r)x3, ẏ = px2y + (r − s)xy2 + sy3, rs(r + s) 6= 0. (3.39)

1.1.2. Possibility s+ v 6= 0 and 2p+ 3r + s+ v = 0. Then v = −(2p+ 3r + s)
and calculations yield U2 = 0. So we arrive at the family of systems

ẋ =(p+ r)x3 − (2p+ 3r)x2y + (p+ r − s)xy2,
ẏ =px2y − (2p+ 2r + s)xy2 + (p+ r)y3.

(3.40)



34 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

These systems, via the transformation

x1 = −r
s

(x− y), y1 =
r

s2
(rx+ sy), t1 =

s3t

r2(r + s)
,

could be brought to the systems

ẋ1 = (p1 + r1)x31, ẏ1 = p1x
2
1y1 + (r1 − s1)x1y

2
1 + s1y

3
1 , (3.41)

where

p1 = −(pr + r2 + ps+ 2rs)/s, r1 = r, s1 = s ⇒ r1s1(r1 + s1) = rs(r + s) 6= 0.

In other words we arrive at systems (3.39).

1.1.3. Possibility (s + v)(2p + 3r + s + v) 6= 0 and 2ps + 2rs − rv = 0. In this
case we have v = 2(ps + rs)/r (this implies q = s(ps − r2 + rs)/r2) and we again
get U2 = 0. This leads to the family of systems

ẋ =(p+ r)x3 + s(2p+ 3r)x2y/r − s(r2 − ps− rs)xy2/r2,
ẏ =px2y + (r2 + 2ps+ 2rs)xy2/r + (p+ r)s2y3/r2.

(3.42)

We claim that these systems could be brought to system (3.39) via a linear trans-
formation and a time rescaling. Indeed, since rs(r + s) 6= 0 we can apply the
change

x1 = rx+ sy, y1 = sy, t1 = t/(rs)

and we obtain systems (3.41), where

p1 = ps/r, r1 = s, s1 = −(r + s) ⇒ r1s1(r1 + s1) = rs(r + s) 6= 0.

So we arrive at systems (3.39) and our claim is proved.

1.2. Subcase 2p + 3r = 0. We have p = −3r/2 and then we obtain U20 =
−9r2(s + v)2/4. Since r 6= 0 the condition U20 = 0 implies v = −s and then we
calculate

U24 = −3q(2q + r + 2s)(2qr + 2rs+ s2)/2 = 0.

So we consider the three possibilities provided by this condition.

1.2.1. Possibility q = 0. Then we obtain U2 = 0 = V̂4 and we arrive at the
family of systems

ẋ = −rx3/2, ẏ = −3rx2y/2 + (r − s)xy2 + sy3,

which is a subfamily of systems (3.39) defined by the condition p = −3r/2.

1.2.2. Possibility q 6= 0 and 2q + r + 2s = 0. In this case q = −(r + 2s)/2 and
we obtain the family of systems

ẋ = −rx3/2− (r + 2s)xy2/2, ẏ = −3rx2y/2 + (r − s)xy2 + sy3,

which is a subfamily of systems (3.40) defined by the condition p = −3r/2 and
which could be brought via an affine transformation to systems (3.39).

1.2.3. Possibility q(2q + r + 2s) 6= 0 and 2qr + 2rs + s2 = 0. In this case we
obtain q = −s(2r + s)/(2r) and we arrive at the family of systems

ẋ =− rx3/2− s(2r + s)xy2/(2r),

ẏ =− 3rx2y/2 + (r − s)xy2 − s2y3/(2r).
It remains to observe that this family of systems is a subfamily of systems (3.42)
defined by the condition p = −3r/2.

Thus the case p 6= 0 is completely investigated.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 35

2. Case p = 0. Then we calculate

U21 = 3r(r + v)(3qr + rs− s2 + rv − sv) = 3rV̂4.

So because of r 6= 0 the condition U21 = 0 implies V̂4 = 0. Since we have to impose
the condition U21 = 0 to be satisfied we examine two subcases: r + v = 0 and
r + v 6= 0.

2.1. Subcase r + v = 0. Hence we have p = 0 and v = −r (this implies U2 = 0)
and we obtain the system

ẋ = rx3 + (s− r)x2y + qxy2, ẏ = (q + s)y3. (3.43)

Because of rs 6= 0 we can apply the transformation

x1 = y, y1 = −rx/s, t1 = st/r

and we obtain systems (3.41) with

p1 = qr/s, r1 = r, s1 = r ⇒ r1s1(r1 + s1) = rs(r + s) 6= 0.

So by an additional transformation we arrive at systems (3.39) again.

2.2. Subcase r+v 6= 0. Then the condition U21 = 0 implies 3qr+rs−s2+rv−sv =

0 and we obtain q = − (r−s)(s+v)
3r . Then calculations yield

U22 = (2s− v)(r + v)(s+ v)(3r + s+ v) = 0

and since r + v 6= 0 we have three possibilities.

2.2.1. Possibility 2s − v = 0. Then v = 2s (this implies q = −s(r − s)/r) and
considering the condition p = 0 we arrive at the family of systems

ẋ = rx3 + 3sx2y − s(r − s)xy2/r, ẏ = (r + 2s)xy2 + s2y3/r.

We observe that this family of systems is a subfamily of systems (3.42) defined by
the condition p = 0, and therefore could be brought to systems of the form (3.39)
via a linear transformation and time rescaling.

2.2.2. Possibility s+ v = 0. Then v = −s and this implies q = 0. Evidently for
p = q = 0 and v = −s system (3.38) becomes a subfamily of systems (3.39) defined
by the condition p = 0.

2.2.3. Possibility 3r + s + v = 0. In this case we have v = −(3r + s) (which
implies q = −(r − s)2/(3r)) and considering the condition p = 0 we obtain the
family of systems

ẋ = rx3 − 3rx2y + (r − s)xy2, ẏ = −(2r + s)xy2 + ry3

which evidently is a subfamily of the family (3.40) defined by the condition p = 0.
So all the cases are examined and we arrive at the following remark.

Remark 3.8. Consider a homogeneous system (3.38) for which the condition U2 =
0 is satisfied. Then this system could be brought via a linear transformation and a
time rescaling to the form (3.39).

Consider the generic cubic system (3.1) with cubic homogeneities as indicated in
system (3.39). Since s 6= 0 via a translation we may assume n = 0 in system (3.1).
Moreover we may assume s = 1 due to the time rescaling t → t/s. So we obtain
the following cubic system

ẋ = a+ cx+ dy + gx2 + 2hxy + ky2 + (p+ r)x3,

ẏ = b+ ex+ fy + lx2 + 2mxy + px2y + (r − 1)xy2 + y3, r(r + 1) 6= 0
(3.44)



36 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

which we consider here below.

3.3.2. Construction of the cubic system possessing invariant lines with configuration
of type T = (3, 1, 1, 1). In what follows we shall determine necessary and sufficient
conditions for a system (3.44) to have a configuration of the type (3, 1, 1, 1).

Considering Remark 2.13 for the homogeneous system (3.39) (with s = 1), asso-
ciated to the system (3.44) we calculate

H(X,Y, Z) = gcd(G1,G2,G3) = −(p+ r)X3(X − Y )Y (rX + Y ).

So each one of the invariant lines y = 0 or y = x or y = −rx of system (3.39) is
simple, whereas the invariant line x = 0 is of multiplicity three.

Remark 3.9. We observe that due to a rescaling, the directions y = −rx and
y = x can be interchanged without changing the form of the system (3.44).

First of all we recall that the directions for the potential invariant lines of system
(3.44) are defined by the factors of the invariant polynomial C3(x, y) = xy(x −
y)(rx+y). Since all non-zero coefficients of the polynomials of degree less than three
of the right-hand parts of system (3.44) are free parameters, we can assume that any
rescaling does not affect them. So we consider only the cubic homogeneous system
associated to the system (3.44). Due to r 6= 0 applying the rescaling (x, y, t) 7→
(x1,−ry1, t1/r2) we obtain the homogeneous cubic system

ẋ1 = (p1 + r1)x31, ẏ1 = p1x
2y1 + (r1 − 1)x1y

2
1 + y31 ,

where p1 = p/r2 and r1 = 1/r. Since x1 = x and y1 = −y/r for these systems we
have

C3(x1, y1) = x1y1(x1 − y1)(r1x1 + y1) = x(−y/r)(x+ y/r)(x/r − y/r)

= − 1

r3
xy(rx+ y)(x− y).

So we conclude that the direction rx+y = 0 (respectively x−y = 0) corresponding
to systems (3.44) passes to the direction x1 − y1 = 0 (respectively rx1 + y1 = 0)
corresponding to the above systems and hence the claim of Remark 3.9 is valid.

Considering Remark 3.9 we examine three possibilities for the direction of a
triplet of parallel invariant lines for systems (3.44): (i) direction x = 0, (ii) direction
y = 0, and (iii) direction y = x.
Existence of a triplet in the direction x = 0. Consider system (3.44). Taking into
account the equations (2.5) and Remark 2.12 for the direction x = 0 we obtain:

Eq7 = k, Eq9 = d− 2hW, Eq10 = a+ gW 2 − cW − (p+ r)W 3.

Evidently we have a triplet of parallel invariant lines in this direction if and only if
the condition k = d = h = 0 holds.

Next we ask for the existence of exactly one invariant line in each one of the
remaining three directions.

1. Direction y = 0. In this case considering the equations (2.5) and Remark
2.12 for systems (3.44) we have

Eq′5 = l − pW, Eq′8 = e− 2mW + (r − 1)W 2, Eq′10 = b− fW −W 3. (3.45)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 37

2. Direction y = x. Considering the equations (2.5) and Remark 2.12 in this
case we calculate

Eq′′6 = l − g − 2h− k + 2m− (1 + p+ 2r)W,

Eq′′9 = −c− d+ e+ f + (l − g + k)W + (1− p− r)W 2,

Eq′′10 = −a+ b+ (e− c)W + (l − g)W 2 − (p+ r)W 3.

(3.46)

3. Direction y = −rx. In this case considering the equations (2.5) and Remark
2.12 we have

Eq′′′6 = 2m− g + 2hr − kr2 − l/r + (p+ 2r + r2)W/r,

Eq′′′9 = f − c− e/r + (l + gr − kr3)W/r2 − (p+ r − r2)W 2/r2,

Eq′′′10 = b+ ar − (e+ cr)W/r + (l + gr)W 2/r2 − (p+ r)W 3/r2.

(3.47)

Thus in what follows we have to impose the existence of exactly one invariant
line in each one of the directions y = 0, y = x and y = −rx providing that the
condition k = d = h = 0 is satisfied.

We observe, that the equation Eq′5 = 0 (respectively Eq′′6 = 0; Eq′′′6 = 0) depends
on the parameter W (linearly) if and only if p 6= 0 (respectively (1 + p + 2r) 6= 0;
(p + 2r + r2) 6= 0). So we consider two cases: p(1 + p + 2r)(p + 2r + r2) 6= 0 and
p(1 + p+ 2r)(p+ 2r + r2) = 0.
Case p(1 + p+ 2r)(p+ 2r+ r2) 6= 0. Considering (3.45) the condition Eq′5 = 0 gives
W = l/p and then we calculate

Eq′8 =
[
ep2 − 2lmp+ l2(r − 1)

]
/p2 = 0, Eq′10 = (bp3 − l3 − lfp2)/p3 = 0.

Since p 6= 0 we obtain the relations e = l(l + 2mp− lr)/p2 and b = l(l2 + fp2)/p3.
These conditions guarantee the existence of an invariant line in the direction y = 0.

Next we consider equations (3.46) associated to the direction y = x providing
that the conditions

k = d = h = 0, e = l(l + 2mp− lr)/p2, b = l(l2 + fp2)/p3 (3.48)

hold. So considering these conditions the equation Eq′′6 = 0 gives W = (l − g +
2m)/(1 + p+ 2r) ≡W ′′0 and then the equation Eq′′9 |{W=W ′′

0 } = 0 yields

f = c− l(l + 2mp− lr)
p2

− (l − g)2 − 4m2

2(1 + p+ 2r)
+

(p− 3)(l − g + 2m)2

2(1 + p+ 2r)2
. (3.49)

Considering this value of the parameter f and (3.48) the equation Eq′′10|{W=W ′′
0 } = 0

implies

a =
l + gp− 2mp+ 2lr

p3(1 + p+ 2r)3

[
cp2(1 + p+ 2r)2 + (l + gp− 2mp+ 2lr)

[
l(p+ r)(1 + 2r)

− pg(1 + r) + 2mp(p+ r)
]]
.

(3.50)
Thus conditions (3.48), (3.49) and (3.50) guarantee the existence of 5 affine

invariant lines of system (3.44): a triplet in the direction x = 0, one line in the
direction y = 0 and one in the direction y = x. It remains now to find the additional
conditions for the existence of one common solution in W of equations Eq′′′6 = 0,
Eq′′′9 = 0 and Eq′′′10 = 0 given in (3.47), providing that the conditions (3.48), (3.49)
and (3.50) hold.



38 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Equation Eq′′′6 = 0 gives W = (l + gr − 2mr)/(p+ 2r + r2) ≡W ′′′0 and then we
calculate

Eq′′′9 |{W=W ′′′
0 } =

(1 + r)G1G2

p2(1 + p+ 2r)2(p+ 2r + r2)2
,

Eq′′′10|{W=W ′′′
0 } = − (1 + r)G1G3

p3(1 + p+ 2r)3(p+ 2r + r2)3
,

where

G1(l, g,m, p, r) = p(g − 2m)(1− r)− l(2 + 3p+ 5r + 2r2) (3.51)

and G2(l, g,m, p, r) and G3(c, l, g,m, p, r) are the following polynomials:

G2 =2mp(p+ r)(1 + 3p+ 7r + r2)− l(−1 + r)(2 + r)(p+ r)(1 + 2r)

− gp (2p2 − r + 6pr + 2r2 − r3),

G3 =cp2r(1 + p+ 2r)2(p+ 2r + r2)2 + l2(p+ r)
[
r3(2 + r)2(1 + 2r)2

+ p(2 + r)(1 + 2r)(−1 + 3r2 + 4r3) + p2(−2 + 6r2 + 5r3)
]

− lp
[
2m(p+ r)(p+ 4p2 + 3p3 − 2r + 8pr + 16p2r − r2 + 30pr2

+ 7p2r2 + 20r3 + 32pr3 + 38r4 + 10pr4 + 22r5 + 4r6)− g(2p3

+ 2p4 − 3pr + 3p2r + 12p3r − 2r2 − 6pr2 + 24p2r2 + 4p3r2 − 5r3

+ 16pr3 + 21p2r3 + 30pr4 + 6p2r4 + 5r5 + 15pr5 + 2r6 + 2pr6)
]

+ (g − 2m)p2r
[
− 2m(p+ r)(1 + 3p+ 3p2 + 6r + 12pr + 13r2

+ 3pr2 + 6r3 + r4) + g(p3 − r − 3pr + 3p2r − 7r2 − 3pr2 − 11r3

− 3pr3 − 7r4 − r5)
]

≡cp2r(1 + p+ 2r)2(p+ 2r + r2)2 + Ψ(l, g,m, p, r).

(3.52)

Therefore we conclude that an invariant line exists in the direction y = −rx if and
only if either G1 = 0, or G2 = G3 = 0. So we examine two subcases: (i) G1 = 0,
(ii) G1 6= 0 and G2 = G3 = 0.

1. Subcase G1 = 0. We observe that the polynomial G1 is linear with respect
to the parameter g with the coefficient p(1 − r) and since p 6= 0 we consider two
possibilities 1− r 6= 0 and 1− r = 0.

1.1. Possibility 1− r 6= 0. Then the equation G1 = 0 yields

g =
2mp(r − 1)− l(2 + 3p+ 5r + 2r2)

p(r − 1)
. (3.53)

It can be checked directly that this condition together with (3.48), (3.49) and

(3.50) lead to system (3.44) which has the invariant lines L̃i = 0 i = i, . . . , 6 with
configuration of type T = (3, 1, 1, 1), where

L̃1 = p(r − 1)x− 3l, L̃2 = p(r − 1)(x− y)− l(2 + r),

L̃3,4 = p2(r − 1)2(p+ r)x2 − 2p(r − 1)
[
l(1 + r + r2)−mp(r − 1)

]
x

+
[
cp2(r − 1)2 − 6l2(1 + r + r2) + 6lmp(r − 1)

]
,

L̃5 = py + l, L̃6 = p(r − 1)(rx+ y)− l(1 + 2r).

(3.54)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 39

To determine if the invariant lines L̃3,4 = 0 are real or complex as well as if the

invariant line L̃1 = 0 coincides with one of the lines L̃3,4 = 0 we calculate:

Discrim[L̃3,4, x] = 4p2(r − 1)2λ(c, l,m, p, r),

Resx(L̃1, L̃3,4) = p2(r − 1)2µ(c, l,m, p, r),

where

λ =
[
mp(r − 1)− l(1 + r + r2)

][
mp(r − 1)− l(1 + 6p+ 7r + r2)

]
− cp2(r − 1)2(p+ r),

µ = cp2(r − 1)2 + 12lmp(r − 1) + 3l2(3p− 4− r − 4r2)

(3.55)

We observe that

sign
(

Discrim[L̃3,4, x]
)

= sign(λ),

i.e. the invariant lines L̃3,4 = 0 are real (respectively complex; coinciding) if λ > 0

(respectively λ < 0; λ = 0). And the invariant line L̃1 = 0 coincides with one of

the lines L̃3,4 = 0 if and only if µ = 0.
On the other hand we observe that the equation λ = 0 is linear with respect to

the parameter c with the coefficient p2(r − 1)2(p + r) and since p(r − 1) 6= 0 we
examine two cases: p+ r 6= 0 and p+ r = 0.

1.1.1. Case p+ r 6= 0. In what follows we examine the possibilities provided by
the polynomials λ and µ.

1.1.1.1. Subcase λ > 0. Then we use a new parameter u setting λ = u2 we
obtain

c =

[
mp(r − 1)− l(1 + r + r2)

][
mp(r − 1)− l(1 + 6p+ 7r + r2)

]
− u2

p2(r − 1)2(p+ r)
. (3.56)

This leads to the system

ẋ =(p+ r)
[
x− 3l

p(r − 1))

][
x+

mp(r − 1)− l(1 + r + r2) + u

p(r − 1)(p+ r)

]
×
[
x+

mp(r − 1)− l(1 + r + r2)− u
p(r − 1)(p+ r)

]
,

ẏ =(y + l/p)
[ l2(r2 − 1)2 − 2lmp(r − 1)(r2 + r + 1)− u2

p2(r − 1)2(p+ r)
+

m2

p+ r

− l2(r + 2)(2r + 1)

p(r − 1)2(p+ r)
+
l + 2mp− lr

p
x− l

p
y + px2 + y2 + (r − 1)xy

]
(3.57)

which has six real invariant lines.
On the other hand for the value of c given in (3.56) we calculate

µ =
1

p+ r
(γ2 − u2), γ = mp(r − 1) + l

[
3p− (r − 1)2

]
(3.58)

and since the condition µ = 0 leads to the coincidence of two invariant lines of the
triplet, we examine two possibilities: µ 6= 0 and µ = 0.

1.1.1.1.1. Possibility µ 6= 0. Then (γ−u)(γ+u) 6= 0 and via the transformation

x1 = αx− 3l(p+ r)

γ − u
, y1 = αy +

l(r − 1)(p+ r)

γ − u
,



40 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

t1 =
(γ − u)2

p2(r − 1)2(p+ r)2
t, α =

p(−1 + r)(p+ r)

γ − u
,

system (3.57) could be brought to the following family of systems (we keep the old
notation for the variables):

ẋ =(p+ r)x(x− 1)(x− v),

ẏ =y
[
(p+ r)v − (p+ r)(1 + v)x+ px2 + (r − 1)xy + y2

]
,

(3.59)

where v = (γ + u)/(γ − u) 6= 0. Moreover v 6= 1 because v − 1 = 2u/(γ − u) 6= 0.
So, considering the conditions on the parameters p and r, which we impose in order
to obtain the above systems, we conclude that for systems (3.59) the following
conditions must hold:

p(1 + p+ 2r)(p+ 2r + r2)r(r2 − 1)v(v − 1) 6= 0. (3.60)

We detect that systems (3.59) have six distinct invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : x = v, L4 : y = 0, L5 : y = x, L6 : rx+ y = 0

and the following nine finite singularities:

M1(0, 0), M2,3

(
0,±

√
−(p+ r)v

)
, M4(1, 0), M5(1, 1), M6(1,−r),

M7(v, 0), M8(v,−rv), M9(v, v).
(3.61)

We observe that the singular points M2 and M3 could be real (if v(p + r) < 0) or
complex (if v(p+ r) > 0), but they could not coincide due to v(p+ r) 6= 0.

Assume first v(p + r) < 0. Then we use a new parameter w setting v(p + r) =
−w2 < 0. We obtain p = −r−w2/v and then the singular points M2 and M3 from
(3.61) become M2,3(0,±w). In this case the condition (3.60) becomes

r(1 + r)vw(v − 1)(v + rv − w2)(rv + r2v − w2)(rv + w2) 6= 0

and we observe that due to r(1 + r)v(v − 1) 6= 0 all the finite singularities are
distinct.

We observe that all the singularities (3.61) are located at the intersection of the
invariant lines, except for M2,3(0,±w) which lie on the line x = 0 and are symmetric
with respect to the origin of coordinates. Since w 6= 0 we deduce that fixing the
position of all invariant straight lines and moving only the singularities M2,3(0,±w)
we could not obtain new configurations. So the distinct configurations depend only
on the position of the invariant lines.

We remark that only two lines are not fixed: L3 : x = v and L6 : y = −rx.
Moreover four of them (namely L1,L4, L5 and L6) intersect at the same point
M1(0, 0). Since this point lies on the invariant line L1, considering the triplet
of parallel invariant lines (L1, L2 and L3) we deduce that we could get different
configurations if L3 is located on the right of L1 (if v > 0) or on the left of it (if
v < 0).

In the case v(p+ r) > 0 we obtain complex singularities M2,3(0,±
√
−(p+ r)v)

whose location does not change the configuration, i.e. the arguments we underlined
above are valid in this case too. So we arrive at the following result.

Lemma 3.10. The family of systems (3.59) with the conditions (3.60) possesses
the following configurations of invariant lines when the corresponding conditions
indicated below are satisfied (examples are given in the last column):

Config. 7.41 ⇔ v(p+ r) < 0, v < 0, (p = 5/2, r = −2, v = −2);



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 41

Config. 7.42 ⇔ v(p+ r) < 0, v > 0, (p = 3/2, r = −2, v = 2);

Config. 7.43 ⇔ v(p+ r) > 0, v < 0, (p = 3/2, r = −2, v = −2);

Config. 7.44 ⇔ v(p+ r) > 0, v > 0, (p = 5/2, r = −2, v = 2);

1.1.1.1.2. Possibility µ = 0. Then (γ − u)(γ + u) = 0 and we may assume
γ + u = 0 because of the change u→ −u. So setting u = −γ 6= 0 in system (3.57)
and applying the transformation

x1 = αx+
3l(p+ r)

2γ
, y1 = αy − l(r − 1)(p+ r)

2γ
,

t1 =
4γ2

p2(r − 1)2(p+ r)2
t, α = −p(r − 1)(p+ r)

2γ

we obtain the 2-parameter family of systems

ẋ = (p+ r)x2(x− 1), ẏ = y
[
− (p+ r)x+ px2 + (r − 1)xy + y2

]
. (3.62)

We observe that this family of systems is a subfamily of (3.59) defined by the
condition v = 0. So considering invariant lines of the system (3.59) for v = 0 we
obtain

L1,3 : x = 0, L2 : x = 1, L4 : y = 0, L5 : y = x, L6 : rx+ y = 0.

Moreover examining the singularities (3.61) we observe that 6 of them coalesced.
More exactly the singular points M2,3, M7, M8 and M9 coalesced with M1(0, 0)
and we obtain a singularity of multiplicity six. As a result we only obtain one
configuration given by Config. 7.45 (providing that the condition p + r 6= 0 is
satisfied).

1.1.1.2. Subcase λ < 0. This means that two invariant lines in the triplet in the
direction x = 0 are complex. We set λ = −u2 < 0 and we obtain

c =
1

p2(r − 1)2(p+ r)

{[
mp(r−1)−l(1+r+r2)

][
mp(r−1)−l(1+6p+7r+r2)

]
+u2

}
.

This leads to the following system

ẋ =− 3l + p(1− r)x
p3(r − 1)3(p+ r)

[[
p(r − 1)(p+ r)x− 3l(p+ r) + γ

]2
+ u2

]
,

ẏ =(y + l/p)
[ l2(r2 − 1)2 − 2lmp(r − 1)(r2 + r + 1) + u2

p2(r − 1)2(p+ r)
+

m2

p+ r

− l2(r + 2)(2r + 1)

p(r − 1)2(p+ r)
+
l + 2mp− lr

p
x− l

p
y + px2 + y2 + (r − 1)xy

] (3.63)

and we consider two possibilities: γ 6= 0 and γ = 0 (γ is given in (3.58)).

1.1.1.2.1. Possibility γ 6= 0. Then via the transformation

x1 = αx− 3l(p+ r)

γ
, y1 = αy − l(r − 1)(p+ r)

γ
,

t1 =
γ2

p2(r − 1)2(p+ r)2
t, α =

p(r − 1)(p+ r)

γ



42 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

system (3.63) could be brought to the following family of systems (we keep the old
notations for the variables):

ẋ =(p+ r)x
[
(x+ 1)2 + v2

]
,

ẏ =y
[
(p+ r)(1 + v2) + 2(p+ r)x+ px2 + (r − 1)xy + y2

]
,

(3.64)

where v = u/γ 6= 0. So, considering the conditions on the parameters p and r which
we impose in order to obtain the above system, we conclude that for the system
(3.59) the following conditions must hold:

p(1 + p+ 2r)(p+ 2r + r2)r(r + 1)(p+ r)v 6= 0. (3.65)

We detect that system (3.64) has six distinct invariant affine straight lines

L1 : x = 0, L2,3 : x = −1± iv, L4 : y = 0, L5 : y = x, L6 : rx+ y = 0

and the following nine finite singularities:

M1(0, 0), M2,3(0,±
√
−(p+ r)(1 + v2)), M4,5(−1± iv, 0),

M6,7(−1± iv,−1± iv), M8,9

(
− 1± iv, (1∓ i)r

)
.

(3.66)

We observe that the singular points M2 and M3 could be real (if p + r < 0) or
complex (if p+ r > 0), but they could not coincide due to p+ r 6= 0.

Assume first p + r < 0. Then we may use a new parameter w setting p + r =
−w2 < 0. So we have p = −r−w2 and then the above singular points M2 and M3

become M2,3(0,±
√

1 + v2 w).
We observe that all the singularities (3.66) are located at the intersections of the

invariant lines, except for M2,3(0,±
√

1 + v2 w) which lie on the line x = 0 and are
symmetric with respect to the origin of coordinates. Since w 6= 0 we deduce that
fixing the position of all the invariant straight lines and moving only the singularities
M2,3(0,±

√
1 + v2 w) we could not obtain new configurations.

On the other hand we remark that only three lines are not fixed: the complex
lines L2,3 : x = −1± iv and L6 : y = −rx. Moreover four of them (namely L1,L4,
L5 and L6) intersect at the same point M1(0, 0). In addition we point out that
according to Notation 1.11, in the triplet of parallel invariant lines L1 (real) and
L2,3 (complex) we place both complex invariant lines on one side of the real line.

We also observe that we have two geometrically distinct positions of the invariant
line L6 : y = −rx: for r < 0 and for r > 0, however this does not lead to distinct
configurations.

When p + r > 0 we obtain complex singularities M2,3(0,±
√
−(p+ r)(1 + v2) )

whose location does not change the configuration, i.e. the arguments mentioned
above are also valid in this case. So we arrive at the following result.

Lemma 3.11. The family of systems (3.64) with the conditions (3.65) possesses
the following configurations of invariant lines when the corresponding conditions
indicated below are satisfied:

Config. 7.46 ⇔ p+ r < 0, (p = 1, r = −2, v = 1);

Config. 7.47 ⇔ p+ r > 0, (p = 3, r = −2, v = 1);



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 43

1.1.1.2.2. Possibility γ = 0. Taking into account (3.58) we obtain m = −l(3p−
1 + 2r − r2)/(p(r − 1)) and then system (3.63) becomes the system

ẋ =− 3l + p(1− r)x
p3(r − 1)3(p+ r)

[[
p(r − 1)(p+ r)x− 3l(p+ r)

]2
+ u2

]
,

ẏ = (y + l/p)
[ l2(p+ r)(9p− 2 + 4r − 2r2) + u2

p2(r − 1)2(p+ r)
+
l
[
(r − 1)2 − 6p

]
p(r − 1)

x

− l

p
y + px2 + y2 + (r − 1)xy

]
.

(3.67)

Since up(p+ r)(r − 1) 6= 0 we can apply the transformation

x1 = αx− 3l(p+ r)

u
, y1 = αy +

l(r − 1)(p+ r)

u
,

t1 =
u2

p2(r − 1)2(p+ r)2
t, α =

p(r − 1)(p+ r)

u

which brings systems (3.67) to the following family of systems (we keep the old
notations for the variables):

ẋ = (p+ r)x(x2 + 1), ẏ = y
[
p+ r + px2 + (r − 1)xy + y2

]
. (3.68)

These systems possess the invariant lines

L1 : x = 0, L2,3 : x = ±i, L4 : y = 0, L5 : y = x, L6 : rx+ y = 0

and the following nine finite singularities:

M1(0, 0), M2,3(0,±
√
−(p+ r) ), M4,5(±i, 0), M6,7(±i,±i), M8,9(±i,∓ir).

We observe that all invariant lines are fixed except for the real line L6 : rx+ y = 0.
In addition we point out that according to Notation 1.11 in the triplet of parallel
invariant lines L1 (real) and L2,3 (complex) we place the real line between the two
complex lines.

As in the previous case, the singularities M2,3(0,±
√
−(p+ r) ) are real if p+r <

0 and they are complex if p + r > 0. So applying the same arguments as in the
previous case, we arrive at the next result.

Lemma 3.12. The family of systems (3.68) with the condition (3.65) (removing
the parameter v which is not relevant here) possesses the following configurations
of invariant lines when the corresponding conditions indicated below are satisfied:

Config. 7.48 ⇔ p+ r < 0, (p = 1, r = −2);

Config. 7.49 ⇔ p+ r > 0, (p = 3, r = −2).

1.1.1.3. Subcase λ = 0. Considering (3.55) and solving the equation λ = 0 with
respect to the parameter c, it is clear that we obtain (3.56) for u = 0. This leads
to systems (3.57) with u = 0, which we denote by (3.57){u=0}.

We observe that in this case we obtain µ = γ2/(p + r) and we again consider
two subcases: µ 6= 0 and µ = 0.

1.1.1.3.1. Possibility µ 6= 0. Then γ 6= 0 and via the transformation

x1 = αx− 3l(p+ r)− γ
γ

, y1 = αy +
l(r − 1)(p+ r)

γ
,

t1 =
γ2

p2(r − 1)2(p+ r)2
t, α =

p(r − 1)(p+ r)

γ



44 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

system (3.57){u=0} can be brought to the following family of systems (we keep the
old notations for the variables):

ẋ =(p+ r)x2(x− 1),

ẏ =y
[
− r + 2rx+ px2 + (1− r)y + (r − 1)xy + y2

]
,

(3.69)

for which the condition p(1 + p+ 2r)(p+ 2r+ r2)r(r+ 1) 6= 0 holds. These systems
possess the following five distinct invariant affine straight lines:

L1,2 : x = 0, L3 : x = 1, L4 : y = 0, L5 : y = x− 1, L6 : rx+ y = r

and six finite singularities

M1(0, 0), M2(0,−1), M3(0, r), M4(1, 0), M5,6

(
1,±

√
−(p+ r)

)
. (3.70)

We detect that three of the above singularities are double. Indeed considering
Lemma 2.7 for system (3.69) we calculate

µ9 = µ8 = 0, µ7 = r2(p+ r)3x4y(px2 + rx2 + y2) 6= 0

i.e. by this lemma M1(0, 0) is a double singular point. Moving M2 (respectively
M3) to the origin of coordinates we obtain for M2:

µ9 = µ8 = 0, µ7 = −(1 + r)2(p+ r)3x4(x− y)(x2 + px2 + rx2 − 2xy + y2) 6= 0;

for M3:

µ9 = µ8 = 0, µ7 = r2(1+r)2(p+r)3x4(rx+y)(px2 +rx2 +r2x2 +2rxy+y2) 6= 0.

So by Lemma 2.7 each one of these singularities is also double.
We observe that the invariant line x = 0 is double and only the line L6 is not

fixed (it depends on the parameter r). Moreover four of the invariant lines (namely
L3, L4, L5 and L6) intersect at the same singular point M4(1, 0).

We notice that the singular points M5 and M6 could be real (if p + r < 0) or
complex (if p+ r > 0), but they could not coincide due to p+ r 6= 0.

Assume first p + r < 0. As before, we may use a new parameter w setting
p+ r = −w2 < 0. So we have p = −r − w2 and then the above singular points M5

and M6 become M5,6(1,±w).
We observe that all the singularities (3.70) are located at the intersections of

the invariant lines, except for M5,6(1,±w) which lie on the line x = 1 and are
symmetric with to the singular point M4(1, 0). Since w 6= 0 we deduce that fixing
the position of all the invariant straight lines and moving only the singularities
M5,6 we could not obtain new configurations. We also observe that we have two
geometrically distinct positions of the invariant line L6 : y = −rx: for r < 0 and
for r > 0, however this does not lead to distinct configurations.

In the case p+r > 0 we obtain complex singularities M5,6

(
1,±

√
−(p+ r)

)
whose

location does not change the configuration, i.e. the arguments mentioned above,
are also valid in this case. So we arrive at the next result.

Lemma 3.13. The family of systems (3.69) with the condition (3.65) possesses
the following configurations of invariant lines when the corresponding conditions
indicated below are satisfied:

Config. 7.50 ⇔ p+ r < 0, (p = 1, r = −2);

Config. 7.51 ⇔ p+ r > 0, (p = 3, r = −2).



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 45

1.1.1.3.2. Possibility µ = 0. So we have λ = µ = 0 and this leads to the
existence of a triple invariant line among (3.54). This leads to the system

ẋ =(p+ r)
[
x− 3l

p(r − 1)

]3
,

ẏ =(y + l/p)
[ l2(9p− 2 + 4r − 2r2)

p2(r − 1)2
− l(6p− 1 + 2r − r2)

p(r − 1)
x− l

p
y

+ px2 + y2 + (r − 1)xy
]

which via the translation x1 = x− 3l
p(r−1) , y1 = y+ l/p can be brought to the cubic

homogeneous system

ẋ = (p+ r)x3, ẏ = y
[
px2 + (r − 1)xy + y2

]
. (3.71)

This system has invariant affine lines of total multiplicity six (one triple and three
simple):

L1,2,3 : x = 0, L4 : y = 0, L5 : y = x, L6 : y = −rx
and only the last invariant line is not fixed. Since the above homogeneous system
has a single singularity M1(0, 0) which is of multiplicity 9, we obtain the unique
configuration given by Config. 7.52.

1.1.2. Case p+ r = 0. Hence we have p = −r and this implies

λ =
[
m(r − 1)r + l(1 + r + r2)

]2
= γ2, µ = c(r − 1)2r2 + 12lγ. (3.72)

So we observe that λ ≥ 0 and we consider two subcases: λ 6= 0 and λ = 0.

1.1.2.1. Subcase λ 6= 0. In this case considering the conditions (3.48), (3.49),
(3.50) and (3.53) we arrive at the family of systems

ẋ =
3l − rx+ r2x

r3(r − 1)3
[
c(r − 1)2r2 + 6lγ − 2(r − 1)rγx],

ẏ =(y − l/r)
[c(r − 1)2r2 − l2(2 + 5r + 2r2) + 6lγ

r2(r − 1)2
+
lr − l + 2mr

r
x

+
l

r
y − rx2 + (r − 1)xy + y2

] (3.73)

and we examine two possibilities: µ 6= 0 and µ = 0.

1.1.2.1.1. Possibility µ 6= 0. Since γ 6= 0 (due to λ 6= 0) we may apply the
transformation

x1 = αx+
6lγ

µ
, y1 = αy − 2l(r − 1)γ

µ
,

t1 =
µ2

4(−1 + r)2r2γ2
t, α =

2r(r − 1)γ

µ

bringing systems (3.73) to the 2-parameter family of systems

ẋ = ax(x− 1), ẏ = y
[
a(x− 1)− rx2 + (r − 1)xy + y2

]
, (3.74)

where a = −4γ2/µ 6= 0. For these system we calculate (see the definition of the
polynomial H(X,Y, Z) on the page 13, Notation 2.4):

H(X,Y, Z) = gcd(G1,G2,G3) = aX(X − Y )Y (rX + Y )(X − Z)Z.



46 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

So, by Lemma 2.6 the invariant line Z = 0 at infinity is of multiplicity two and the
invariant affine lines are

L1 : x = 0, L2 : x = 1, L3 : y = 0, L4 : y = x, L5 : y = −rx.

We detect that the above system has six finite singularities

M1(0, 0), M2,3(0,±
√
a), M4(1, 0), M5(1, 1), M6(1,−r),

which are distinct due to a 6= 0. We observe that only the line L5 is not fixed (it
depends on the parameter r). Moreover four of the invariant lines (namely L1, L3,
L4 and L5) intersect at the same singular point M4(1, 0).

On the other hand the singular points M2 and M3 could be real (if a > 0) or
complex (if a < 0), but they could not coincide due to a 6= 0. These two singularities
are located on the invariant line x = 0 and are symmetric with respect to the origin
of coordinates.

So after the examination of the position of the invariant lines we arrive at the
next result.

Lemma 3.14. The family of systems (3.74) with the condition ar(r−1)(r+1) 6= 0
possesses the following configurations of invariant lines when the corresponding
conditions indicated below are satisfied:

Config. 7.53 ⇔ a > 0, (a = 1, r = −2);

Config. 7.54 ⇔ a < 0, (a = −1, r = −2).

1.1.2.1.2. Possibility µ = 0. Considering (3.72) we obtain

µ = c(r − 1)2r2 + 12lγ = 0 ⇒ c = −12lγ/((r − 1)2r2)

and then systems (3.73) with the above value for the parameter c, via the translation

x1 = x+ 3l/(r(1− r)), y1 = u− l/r

will be brought to the family of systems:

ẋ = ax2, ẏ = y
[
ax− rx2 + (r − 1)xy + y2

]
, (3.75)

where a = 2γ/(r(1− r)) 6= 0. For this system we calculate

H(X,Y, Z) = −aX2(X − Y )Y (rX + Y )Z

and therefore by Lemma 2.6 the invariant line at infinity, Z = 0, is of multiplicity
two. This system has four distinct invariant affine lines

L1,2 : x = 0, L3 : y = 0, L4 : y = x, L5 : y = −rx

and the unique finite singularity M1(0, 0). Considering Lemma 2.7 for the above
system we calculate

µ0 = µ1 = µ2 = 0, µ3 = a3(x− y)y(rx+ y) 6= 0,

µ4 = µ5 = µ6 = µ7 = µ8 = µ9 = 0

and hence by this lemma, three finite singularities coalesced with singularities at
infinity (together with one invariant line) and the remaining singularity M1(0, 0) is
of multiplicity six.

Since the line x = 0 is double, it is not too hard to detect that systems (3.75)
have the unique configuration of invariant lines given by Config. 7.55.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 47

1.1.2.2. Subcase λ = 0. Considering (3.72) we obtain γ = 0. So from (r−1) 6= 0
we obtain

γ = m(1− r)r − l(1 + r + r2) = 0 ⇒ m = l(1 + r + r2)/(r(1− r))

and then µ = c(r− 1)2r2 6= 0. In this case the right-hand part of the first equation
of systems (3.73) becomes linear in x and applying the translation

x1 = x+ 3l/(r(1− r)), y1 = u− l/r

we obtain the family of systems:

ẋ = cx, ẏ = y
[
c− rx2 + (r − 1)xy + y2

]
, (3.76)

with cr(r2 − 1) 6= 0. For these systems we calculate

H(X,Y, Z) = cX(X − Y )Y (rX + Y )Z2

and therefore by Lemma 2.6 the invariant line at infinity, Z = 0, is of multiplicity
three. These systems possess the invariant affine lines

L1 : x = 0, L2 : y = 0, L3 : y = x, L4 : y = −rx

and three finite singularities

M1(0, 0), M2,3(0,±
√
−c).

So examining in the same manner as further above the position of the invariant
lines as well as the position of the singularities M2,3(0,±

√
−c ) (which could be

real or complex) we obtain the following two distinct configurations: Config. 7.56
if c < 0 and Config. 7.57 if c > 0.

1.2. Possibility r = 1. Then considering (3.51) we obtain G1 = −3l(3 + p) = 0.
On the other hand the condition 1 + p+ 2r 6= 0 (see the paragraph 3.3.2) for r = 1
gives p+ 3 6= 0. Therefore we obtain l = 0 and considering (3.48), (3.49) and (3.50)
we have the conditions:

k = d = h = e = b = l = 0,

f =
c(3 + p)2 − (g − 2m)(3g + 2mp)

(3 + p)2
,

a =
c(3 + p)2 − 2(g − 2m)(g +m+mp)

(3 + p)3
.

In this case the cubic system (3.44) takes the form

ẋ =
[
x+

g − 2m

3 + p

][
c− 2(g − 2m)(g +m+mp)

(3 + p)2
− 2(g +m+mp)x

3 + p
x+ (1 + p)x2

]
,

ẏ =y
[ (2m− g)(3g + 2mp)

(3 + p)2
+ c+ 2mx+ px2 + y2

]
.

(3.77)
These systems possess the invariant lines

L̃1 = g − 2m+ (3 + p)x, L̃2,3 = c(3 + p)2 − 2(g − 2m)(g +m+mp)

+ 2(3 + p)(g +m+mp)x+ (1 + p)(3 + p)2x2,

L̃4 = y, L̃5 = g − 2m+ (3 + p)(x− y), L̃6 = g − 2m+ (3 + p)(x+ y).,



48 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

To determine if the invariant lines L̃2,3 = 0 are real or complex as well as if the

invariant line L̃1 = 0 coincides with one of the lines L̃2,3 = 0 we calculate:

Discrim[L̃2,3, x] = 4(3 + p)2λ′(c, g,m, p), Resx(L̃1, L̃2,3) = (3 + p)2µ′(c, g,m, p),

where

λ′ = (3g − 3m+ 2gp− 3mp)(g +m+mp)− c(1 + p)(3 + p)2,

µ′ = c(3 + p)2 + (g − 2m)(−3g − 6m+ gp− 6mp).
(3.78)

We observe that

sign
(

Discrim[L̃2,3, x]
)

= sign(λ′),

i.e. the invariant lines L̃2,3 = 0 are real (respectively complex; coinciding) if λ′ > 0

(respectively λ′ < 0; λ′ = 0). On the other hand the invariant line L̃1 = 0 coincides

with one of the lines L̃2,3 = 0 if and only if µ′ = 0.
We observe that the equation λ′ = 0 is linear with respect to the parameter c

with the coefficient (1 + p)(3 + p)2 and since (3 + p) 6= 0 we examine two cases:
p+ 1 6= 0 and p = −1.

1.2.1. Case p+ 1 6= 0. In what follows we examine the possibilities provided by
the polynomials λ′ and µ′.

1.2.1.1. Subcase λ′ > 0. Then we use a new parameter u setting λ′ = u2 and
we obtain

c =
1

(1 + p)(3 + p)2
[
(3g − 3m+ 2gp− 3mp)(g +m+mp)− u2

]
. (3.79)

This leads to the system

ẋ =
1

1 + p

[
x+

g − 2m

3 + p

][g +m+mp− u
3 + p

+ (1 + p)x
]

×
[g +m+mp+ u

3 + p
+ (1 + p)x

]
,

ẏ =y
[m(1 + p)(6g − 3m+mp)− g2p− u2

(1 + p)(3 + p)2
+ 2mx+ px2 + y2

]
.

(3.80)

On the other hand for the value of c given in (3.79) we calculate

µ′ =
1

1 + p
(γ′2 − u2), γ′ = gp− 3m(1 + p) (3.81)

and since the condition µ′ = 0 leads to the merging of two invariant lines of the
triplet, we examine two subcases: µ′ 6= 0 and µ′ = 0.

1.2.1.1.1. Possibility µ′ 6= 0. Then (γ′−u)(γ′+u) 6= 0 and via the transforma-
tion

x1 = αx+
(g − 2m)(1 + p)

γ′ + u
, y1 = αy,

t1 =
(γ′ + u)2

(1 + p)2(3 + p)2
t, α =

(1 + p)(3 + p)

γ′ + u

system (3.57) can be brought to the following family of systems (we keep the old
notations for the variables):

ẋ = (p+ 1)x(x− 1)(x− v), ẏ = y
[
(p+ 1)v − (p+ 1)(1 + v)x+ px2 + y2

]
.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 49

We observe that this family of systems is a subfamily of the family (3.59), defined
by the condition r = 1. According to Lemma 3.10 for systems (3.59) the condition
(3.60) has to be satisfied, including r 6= 1. However we detect that for this family
the hypersurface r = 1 is not part of the bifurcation set of configurations as it can
be seen from Lemma 3.10. So no new configurations can be obtained.

1.2.1.1.2. Possibility µ′ = 0. Then (γ′ − u)(γ′ + u) = 0 and we may assume
γ′+ u = 0 because of the change u→ −u. So setting in system (3.80) u = −γ′ 6= 0
(since λ′ = γ′2 > 0) and applying the transformation

x1 = αx+
(g − 2m)(1 + p)

2γ′
, y1 = αy,

t1 =
4γ′2

(1 + p)2(3 + p)2
t, α =

(1 + p)(3 + p)

2γ′

we obtain the 1-parameter family of systems:

ẋ = (p+ 1)x2(x− 1), ẏ = y
[
− (p+ 1)x+ px2 + y2

]
. (3.82)

We observe that (3.82) is a subfamily of systems (3.62) defined by the condition
r = 1. So we deduce that we do not have new configurations.

1.2.1.2. Subcase λ′ < 0. This means that two invariant lines of the triplet in
the direction x = 0 are complex. We set λ′ = −u2 < 0 and we obtain

c =
1

(1 + p)(3 + p)2
[
(3g − 3m+ 2gp− 3mp)(g +m+mp) + u2

]
. (3.83)

This leads to the systems

ẋ =
1

(1 + p)(3 + p)2

[
x+

g − 2m

3 + p

]
×
[(
g +m+mp+ 3x+ 4px+ p2x

)2
+ u2

]
,

ẏ =y
[m(1 + p)(6g − 3m+mp)− g2p+ u2

(1 + p)(3 + p)2
+ 2mx+ px2 + y2

] (3.84)

We consider two possibilities: γ′ 6= 0 and γ′ = 0.

1.2.1.2.1. Possibility γ′ 6= 0. Then via the transformation

x1 = αx− (g − 2m)(1 + p)

γ′
, y1 = αy,

t1 =
γ′2

(1 + p)2(3 + p)2
t, α = − (1 + p)(3 + p)

γ′

systems (3.84) can be brought to the following family of systems (we keep the old
notations for the variables):

ẋ =(p+ 1)x
[
(x+ 1)2 + v2

]
,

ẏ =y
[
(p+ 1)(1 + v2) + 2(p+ 1)x+ px2 + y2

]
,

(3.85)

where v = u/γ′ 6= 0. It could be easily observed that (3.85) is a subfamily of
systems (3.64) defined by the condition r = 1. This family was investigated earlier
(see Lemma 3.11) and since for r = 1 we do not have bifurcation points for the
family (3.64) we deduce that there are no new configurations.



50 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

1.2.1.2.2. Possibility γ′ = 0. Taking into account (3.81) we obtain m =
gp/(3(1 + p)) and then system (3.84) becomes system

ẋ =
[
(1 + p)x+ g/3

][(
x+

g

3(1 + p

)2
+

u2

(1 + p)2(3 + p)2

]
,

ẏ =y
[g2p(3 + p)2 + 9(1 + p)u2

9(1 + p)2(3 + p)2
+

2gp

3(1 + p)
x+ px2 + y2

]
.

(3.86)

Since u(1 + p)(3 + p) 6= 0 we can apply the transformation

x1 = αx− g(3 + p)

3u
, y1 = αy, t1 =

u2

(1 + p)2(3 + p)2
t, α =

(1 + p)(3 + p)

u

which brings systems (3.86) to the systems

ẋ = (p+ 1)x(x2 + 1), ẏ = y
[
p+ 1 + px2 + y2

]
.

We observe that this family of systems is a subfamily of (3.68) defined by the
condition r = 1. So we again do not have new configurations.

1.2.1.3. Subcase λ′ = 0. Considering (3.78) and solving the equation λ′ = 0
with respect to the parameter c, it is clear that we obtain (3.83) for u = 0. This
leads to the system (3.84) with u = 0, which we denote by (3.84){u=0}.

We observe that in this case we obtain µ′ = γ′ and we again consider two
possibilities: µ′ 6= 0 and µ′ = 0.

1.2.1.3.1. Possibility µ′ 6= 0. Then γ′ 6= 0 and via the transformation

x1 = αx−g +m+mp

γ′
, y1 = αy, t1 =

γ′2

(1 + p)2(3 + p)2
t, α = − (1 + p)(3 + p)

γ′

system (3.84){u=0} could be brought to the system

ẋ = (p+ 1)x2(x− 1), ẏ = y
[
− y + 2xy + px2y + y3

]
.

It remains to observe that we obtain a subfamily of (3.69), defined by the condition
r = 1.

1.2.1.3.2. Possibility µ′ = 0. In this case we have λ′ = µ′ = γ′ = 0 (i.e.
m = gp/(3(1 + p))) and we obtain a triple line in the direction x = 0. Therefore
system (3.84){u=0} leads to the system

ẋ = (1 + p)
[
x+

gp

3(1 + p)

]
, ẏ = y

[
p
(
x+

gp

3(1 + p)

)
+ y2

]
and evidently making the translation x1 = x+ gp

3(1+p) , y1 = y we arrive at the cubic

homogeneous systems

ẋ = (p+ 1)x3, ẏ = y
[
px2 + y2

]
which form a subfamily of systems (3.71) for r = 1.

1.2.2. Case p = −1. In this case system (3.77) becomes

ẋ =
[
x+ (g − 2m)/2

][
c− g(g − 2m)/2 + gx)

]
,

ẏ =y
[
c− (g − 2m)(3g − 2m)/4 + 2mx− x2 + y2

] (3.87)

for which

λ′ = g2, µ′ = 4(c− g2 + 2gm). (3.88)

So we observe that λ′ ≥ 0 and we consider two subcases: λ′ 6= 0 and λ′ = 0.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 51

1.2.2.1. Subcase λ′ 6= 0. Then g 6= 0 and we have to consider two possibilities:
µ′ 6= 0 and µ′ = 0.

1.2.2.1.1. Possibility µ′ 6= 0. In this case systems (3.87) can be brought via the
transformation

x1 = αx+
2g(g − 2m)

µ′
, y1 = αy, t1 =

µ′2

16g2
t, α = −4g

µ′

to the systems
ẋ = ax(x− 1), ẏ = y

[
a(x− 1)− x2 + y2

]
,

where a = −4/µ′. We observe that we obtain a subfamily of systems (3.74) defined
by r = 1.

1.2.2.1.2. Possibility µ′ = 0. Considering (3.88) we obtain c = g(g − 2m) and
then system (3.87) takes the form

ẋ = g(g − 2m+ 2x)2/4, ẏ = y
[
(g2 − 4m2 + 8mx− 4x2 + 4y2)

]
/4.

In this case we apply the transformation x1 = x/g + (g − 2m)/(2g), y1 = y/g,
t1 = g2t which brings the above system to the system

ẋ = x2, y(x− x2 + y2).

Evidently this system belongs to the family (3.75) for a = r = 1, already studied.

1.2.2.2. Subcase λ′ = 0. By (3.88) we obtain g = 0 and system (3.87) becomes

ẋ = c(x−m), y
[
c− (x−m)2 + y2

]
.

Clearly applying the translation x→ x+m we obtain the system

ẋ = cx, y
[
c− x2 + y2

]
which belongs to the family of systems (3.76) for r = 1, already studied.

2. Subcase G1 6= 0, G2 = G3 = 0. Then considering (3.52), since p(1 + p +
2r)(p+2r+r2) 6= 0, the condition G3 = 0 yields c = −Ψ(l, g,m, p, r). On the other
hand we observe that the polynomial G2 from (3.52) is linear with respect to the
parameter m with the coefficient 2p(p + r)w, where w = 1 + 3p + 7r + r2. Since
p 6= 0 it remains to consider three possibilities: (a) (p+ r)w 6= 0, (b) p+ r 6= 0 and
w = 0, and (c) p+ r = 0.

2.1. Possibility (p+ r)w 6= 0. Then the condition G2 = 0 gives

m =
1

2p(p+ r)w

[
l(r − 1)(2 + r)(p+ r)(1 + 2r) + gp(2p2 − r + 6pr + 2r2 − r3)

]
.

It can be checked directly that this condition together with (3.48), (3.49), (3.50)
and c = −Ψ(l, g,m, p, r) applied to systems (3.44) lead to the family of systems:

ẋ = (p+ r)
[
x+
−3l(2 + r)(p+ r) + gp(1 + p+ 2r)

p(p+ r)w

]
×
[
x+
−3l(−1 + r)(p+ r) + gp(p+ 3r)

p(p+ r)w

]
×
[
x+

3l(p+ r)(1 + 2r) + gp(p+ 2r + r2)

p(p+ r)w

]
≡ (p+ r)(x+ ϕ1)(x+ ϕ2)(x+ ϕ3),

ẏ = −(y + l/p)
[ 1

p2(p+ r)2w2

[
(3lg p(r − 1)(p+ r)(2p2 − r + 3pr − r2 − r3)



52 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

− g2p2(p3 + 6p2r − r2 + 9pr2 + 2r3 − r4) + l2(p+ r)2(2 + 3p+ 18p2 + r

+ 21pr − 6r2 + 3pr2 + r3 + 2r4)
]
− 1

p(p+ r)w

[
− l(−1 + r)(p+ r)(−1 + 3p

+ 2r − r2) + gp(2p2 − r + 6pr + 2r2 − r3)
]
x+ ly/p− px2 + (1− r)xy − y2

]
.

We calculate

ϕ2 − ϕ1 =
gp(r − 1) + 9l(p+ r)

p(p+ r)w
, ϕ3 − ϕ1 =

(1 + r)
[
gp(r − 1) + 9l(p+ r)

]
p(p+ r)w

and hence setting δ = gp(r − 1) + 9l(p + r) we deduce that if δ = 0 then all the
parallel invariant lines in the direction x = 0 coincide.

On the other hand for the above mentioned values of the parameters c and d we
obtain

G1 = − (1 + p+ 2r)(p+ 2r + r2)δ

p(p+ r)w

and therefore the condition G1 6= 0 implies δ 6= 0. In this case we can apply the
transformation

x1 = αx+
3l(2 + r)(p+ r)− gp(1 + p+ 2r)

δ
, y1 = αy − l(p+ r)w)

δ
,

t1 =
δ2

p2(p+ r)2w2
t, α = −p(p+ r)w

δ

and we arrive at the family of systems

ẋ = (p+ r)x(x− 1)(x− 1− r),
ẏ = y

[
(1 + r)(p+ r) + (r2 − 2p− r)x+ px2 + (1 + p+ 2r)y

+ (r − 1)xy + y2
]
,

(3.89)

for which the following condition is satisfied

r(r + 1)p(1 + p+ 2r)(p+ 2r + r2)(p+ r)(1 + 3p+ 7r + r2) 6= 0. (3.90)

These systems possess six real invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : x = r + 1,

L4 : y = 0, L5 : y = x− 1− r, L6 : rx+ y = 0

which are all distinct because r(r + 1) 6= 0. The above systems have the following
nine real finite singularities:

M1(0, 0), M2(0,−1− r), M3(0,−p− r), M4(1, 0), M5(1 + r, 0),

M6(1,−r), M7(1,−p− 2r), M8(1 + r,−p− r), M9

(
1 + r,−r(1 + r)

)
.

We observe that because of the condition r(r+ 1)(p+ r) 6= 0 all these singularities
are distinct except for the case of the singularity M7(1,−p − 2r) which coalesces
with the singularity M4 if p+ 2r = 0.

It is easy to determine that 6 of these singularities are located at the intersections
of the above invariant lines, more precisely these are the singular points Mi for
i ∈ {1, 2, 4, 5, 6, 9}. The singular point M3 (respectively M7; M8) is located on the
invariant line L1 (respectively L2; L3). Moreover we have three singular points each
one located at the intersection of three invariant lines: L1, L2 and L6 intersect at
the point M1; L3, L4 and L5 intersect at the point M5; and L2, L5 and L6 intersect
at the point M6.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 53

To determine all possible configurations for system (3.89) we have to examine
the positions of the invariant lines as well as of the singularities M6, M8 and M9

depending on the parameters r and u.
Let us first examine the position of the invariant lines. We observe that three of

the lines are fixed, namely L1, L2 and L4. Other three invariant lines depend on
the parameter r. More exactly the positions of L1 and L5 depend on the sign of the
expression r + 1, whereas the position of the invariant line L6 depends on the sign
of the parameter r. So it is clear that we have to examine three cases: (i) r < −1;
(ii) − 1 < r < 0 and (iii) r > 0.

Next we consider the position of the finite singularityM3(0, y3) with y3 = −(p+r)
(respectively M7(1, y7) with y7 = −(p + 2r); M8(1 + r, y8) with y8 = −r(1 + r))
on the invariant line x = 0 (respectively x = 1; x = 1 + r) with respect to the
singular points M1(0, 0) and M2(0,−(1 + r) (respectively M4(1, 0) and M6(1,−r);
M5(1 + r, 0) and M9(1 + r,−r(1 + r))

)
. It is clear that the positions of these

singularities in three different cases (i) − (iii) enumerated above could be distinct
and therefore we examine each one of these three cases.

Case (i) r < −1. Then considering Notation 3.6 we have

0 < −(r+1) ⇒ M0 ≺M2; 0 < −r ⇒ M4 ≺M6; −r(r+1) < 0 ⇒ M9 ≺M5,

and considering the coordinates y3 = −(p+ r), y7 = −(p+ 2r) and y8 = −r(1 + r)
we have the next implications.

(I) For the singular point M3:

y3 < 0 ⇒ M3 ≺M0 ≺M2; 0 < y3 < −(r + 1) ⇒ M0 ≺M3 ≺M2;

y3 > −(r + 1) ⇒ M0 ≺M2 ≺M3.

(II) For the singular point M7:

y7 ≤ 0 ⇒ M7 �M4 ≺M6; 0 < y7 < −r ⇒ M4 ≺M7 ≺M5;

y7 > −r ⇒ M4 ≺M5 ≺M7.

(III) For the singular point M8:

y8 < −r(r + 1) ⇒ M8 ≺M9 ≺M5; −r(r + 1) < y8 < 0 ⇒ M9 ≺M8 ≺M5;

y8 > 0 ⇒ M9 ≺M5 ≺M8.

Case (ii) −1 < r < 0. In this case we have

−(r + 1) < 0⇒M2 ≺M0; 0 < −r ⇒M4 ≺M6; 0 < −r(r + 1)⇒M5 ≺M9,

and this leads to the next implications.

(I) For the singular point M3:

y3 < −(r + 1) ⇒ M3 ≺M2 ≺M0; −(r + 1) < y3 < 0 ⇒ M2 ≺M3 ≺M0;

y3 > 0 ⇒ M2 ≺M0 ≺M3.

(II) For the singular point M7:

y7 ≤ 0 ⇒ M7 �M4 ≺M6; 0 < y7 < −r ⇒ M4 ≺M7 ≺M6;

y7 > −r ⇒ M4 ≺M6 ≺M7.

(III) For the singular point M8:

y8 < 0 ⇒ M8 ≺M5 ≺M9; 0 < y8 < −r(r + 1) ⇒ M5 ≺M8 ≺M9;



54 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

y8 > −r(r + 1) ⇒ M5 ≺M9 ≺M8.

Case (iii) r > 0. Then we have

−(r+ 1) < 0 ⇒ M2 ≺M0; −r < 0 ⇒ M6 ≺M4; −r(r+ 1) < 0 ⇒ M9 ≺M5,

and this leads to the next implications.

(I) For singular point M3:

y3 < −(r + 1) ⇒ M3 ≺M2 ≺M0; −(r + 1) < y3 < 0 ⇒ M2 ≺M3 ≺M0;

y3 > 0 ⇒ M2 ≺M0 ≺M3.

(II) For singular point M7:

y7 < −r ⇒ M7 ≺M6 ≺M4; −3 < y7 < 0 ⇒ M6 ≺M7 ≺M4;

y7 ≥ 0 ⇒ M6 ≺M4 �M7.

(III) For singular point M8:

y8 < −r(r + 1) ⇒ M8 ≺M9 ≺M5; −r(r + 1) < y8 < 0 ⇒ M9 ≺M8 ≺M5;

y8 > 0 ⇒ M9 ≺M5 ≺M8.

Since we only have two parameters (p and r), clearly not all of the possibilities de-
scribed above could be realizable. So examining the compatibilities of the conditions
it is not too hard to convince ourselves (using for example, the tools “FindInstance”
or “Reduce” of computer algebra system Mathematica) that the following lemma
is valid.

Lemma 3.15. The family of systems (3.89) with the condition (3.90) possesses
the following configurations of invariant lines when the corresponding conditions
indicated below are satisfied:

Config. 7.58 ⇔ p > 1, p > −2r, p > r2, (p = 5, r = −2);

Config. 7.59 ⇔



r < −1,−2r < p < r2, (p = 17/2, r = −3);

or r < −1, r2 < p < −2r, (p = 65/24, r = −3/2);

or − 1 < r < 0,−2r < p < 1, (p = 3/4, r = −1/4);

or − 1 < r < 0, 1 < p < −2r, (p = 131/96, r = −3/4);

or r > 0, 1 < p < r2, (p = 7/2, r = 2);

or r > 0, r2 < p < 1, (p = 5/8, r = 1/2);

Config. 7.61 ⇔;−2 < r < −1/2, p = −2r, (p = 3, r = −3/2);

Config. 7.62 ⇔

{
r < −2, p = −2r, (p = 6, r = −3);

or − 1/2 < r < 0, p = −2r, (p = 3/4, r = −3/8);

Config. 7.64 ⇔


r < −1,−r < p < −2r, p < r2, (p = 5/2, r = −2);

or − 1 < r < 0,−r < p < −2r, p < 1, (p = 7/8, r = −1/2);

or r > 0,−r < p < r2, p < 1, (p = −1/2, r = 1);

Config. 7.65 ⇔


r < −1, 1 < p < −r, (p = 3/2, r = −2);

or − 1 < r < 0, r2 < p < −r, (p = 3/8, r = −1/2);

or r > 0,−2r < p < −r, (p = −1, r = 3/4);



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 55

Config. 7.66 ⇔


r < −1, p < 1, (p = −1/2, r = −2);

or − 1 < r < 0, p < r2, (p = −1/2, r = −1/2);

or r > 0, p < −2r, (p = −7/2, r = 1);

Config. 7.67 ⇔ r > 0, p = −2r, (p = −2, r = 1).

2.2. Possibility (p+ r) 6= 0, w = 0. Then the condition w = 1 + 3p+ 7r+ r2 = 0
gives p = −(1 + 7r + r2)/3 6= 0 (since p 6= 0) and we denote w′ = 1 + 7r + r2 6= 0.
Then we calculate

G2 = (r − 1)(2 + r)(1 + 2r)
[
g(r − 1)w′ + 9l(1 + 4r + r2)

]
/27,

p(1 + p+ 2r)(p+ 2r + r2) = (r − 1)2(2 + r)(1 + 2r)w′/9 6= 0.

Hence the condition G2 = 0 gives

g = −9l(1 + 4r + r2)

(r − 1)w′

and this leads to the family of systems

ẋ =− 1 + 4r + r2

3

[
x−

3
[
2m(1− r)w′ − 3l(2 + r)(1 + 5r)

]
(r − 1)2(2 + r)w′

]
×
[
x− 3(3l(5 + r)(1 + 2r) + 2m(r − 1)w′)

(r − 1)2(1 + 2r)w′

]
×
[
x− 3(3l(2 + r)(1 + 2r) + 2m(r − 1)w′)

(r − 1)(2 + r)(1 + 2r)w′

]
≡ (p+ r)(x− ϕ′1)(x− ϕ′2)(x− ϕ′3),

ẏ =(y − 3l/w′)
[ −3

(r − 1)4(2 + r)2(1 + 2r)2w′
[
4m2(r − 1)2w′(1 + 5r + 15r2

+ 5r3 + r4) + 3l2(2 + r)2(5 + r)(1 + 2r)2(1 + 5r) + 6lm(r − 1)(2 + r)(1 + 2r)

× (2 + 19r + 66r2 + 19r3 + 2r4)
]

+
3l(r − 1) + 2mw′

w′
x+ 3ly/w′ − w′x2/3

+ (r − 1)xy + y2
]
.

We calculate

ϕ′2 − ϕ′1 =
18(1 + r)δ′

(r − 1)2(2 + r)(1 + 2r)w′
, ϕ′3 − ϕ′1 =

18rδ′

(r − 1)2(2 + r)(1 + 2r)w′

where δ′ = 3l(2 + r)(1 + 2r) + m(r − 1)w′. Since r(r + 1) 6= 0 it is clear that
the condition δ′ = 0 is equivalent to the existence of a triple invariant line in the
direction x = 0.

On the other hand for the above mentioned values of the parameters p and g we
obtain G1 = −2δ′/3 6= 0. Therefore we can apply the transformation

x1 = αx−
(r − 1)

[
m(r − 1)w′ + δ′

]
6δ′

, y1 = αy − l(r − 1)2(2 + r)(1 + 2r))

6δ′
,

t1 =
324δ′2

(r − 1)4(2 + r)2(1 + 2r)2w′2
t, α =

(r − 1)2(2 + r)(1 + 2r)w′

18δ′

and we arrive at the family of systems

ẋ =− (1 + 4r + r2)x(x− 1)(x+ r)/3,



56 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

ẏ =y
[
− r(1 + r + r2)− 3(r − 1)rx− (1 + 7r + r2)x2 + (r − 1)2y

+ 3(r − 1)xy + 3y2
]
/3.

We observe that these system via the transformation (x, y, t) 7→ (1−x,−y, t) could
be brought to the system (3.89) with p = −(1 + 7r + r2)/3 6= 0 and hence no new
configurations can be obtained.

2.3. Possibility p+ r = 0. Considering (3.52) for p = −r we obtain

c = g(l − 2lr + gr − 2mr)/r2, G2 = −gr2(1 + r)2

and because of r(r+ 1) 6= 0 the condition G2 = 0 gives g = 0. However in this case
we obtain c = 0 and this leads to degenerate systems

ẋ = 0, ẏ = (y − l/r)
[ 2mr − l
r(1 + r)

+ rx+ y
][ l + 2m

1 + r
− x+ y

]
.

So the case p(1 + p+ 2r)(p+ 2r + r2) 6= 0 is completely examined.
Case p(1 + p+ 2r)(p+ 2r+ r2) = 0. We examine two subcases: p = 0 or p 6= 0 and
(1 + p+ 2r)(p+ 2r + r2) = 0.

1. Subcase p = 0. Considering (3.45) the condition Eq′5 = 0 gives l = 0 and then
we calculate

Eq′5 = 0, Eq′8 = e− 2mW + (r − 1)W 2, Eq′10 = b− fW −W 3. (3.91)

Now taking into consideration the conditions k = d = h = p = l = 0 we consider
other two directions. From (3.46) and (3.47) we obtain respectively:

Eq′′6 = −g + 2m− (1 + 2r)W, Eq′′9 = −c+ e+ f − gW + (1− r)W 2,

Eq′′10 = −a+ b+ (e− c)W − gW 2 − rW 3.
(3.92)

and

Eq′′′6 = 2m− g + (2 + r)W, Eq′′′9 = f − c− e/r + gW/r + (r − 1)W 2/r,

Eq′′′10 = b+ ar − (e+ cr)W/r + gW 2/r −W 3/r.
(3.93)

As we observe the equation Eq′′6 = 0 (respectively Eq′′′6 = 0) is linear in W if and
only if 1 + 2r 6= 0 (respectively 2 + r 6= 0). So in what follows we consider two
possibilities: (1 + 2r)(2 + r) 6= 0 and (1 + 2r)(2 + r) = 0.

1.1. Possibility (1 + 2r)(2 + r) 6= 0. Then the equation Eq′′6 = 0 gives W =
(2m− g)/(1 + 2r) ≡W ′′0 and then the equation Eq′′9 |{W=W ′′

0 } = 0 yields

f = c− e− (g − 2m)(2g − 2m+ gr + 2mr)

(1 + 2r)2
. (3.94)

Considering this value of the parameter f and (3.92) the equation Eq′′10|{W=W ′′
0 } = 0

gives

a = b+
(c− e)(g − 2m)

1 + 2r
− (g − 2m)2(g + gr + 2mr))

(1 + 2r)3
. (3.95)

We examine now the equations corresponding to the direction y = −rx. Consider-
ing (3.93) and the conditions (3.94) and (3.95) we detect that the equation Eq′′′6 = 0
gives W = (g−2m)/(2 + r) ≡W ′′′0 and then the equation Eq′′′9 |{W=W ′′′

0 } = 0 yields

e = − (g − 2m)(r − 1)(g + 2m− 2gr + 14mr + gr2 + 2mr2)

(2 + r)2(1 + 2r)2
. (3.96)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 57

Considering this value of the parameter e and (3.93) the equation Eq′′′10|{W=W ′′′
0 } =

0 implies

b = − (g − 2m)(r − 1)

(2 + r)3(1 + 2r)3
[
c(2 + r)2(1 + 2r)2 − (g − 2m)(8g − 4m+ 11gr

+ 8mr + 8gr2 − 4mr2)
]
.

(3.97)

Now we return to the direction y = 0. Taking into account (3.91) and the conditions
(3.94), (3.95), (3.96) and (3.97) we obtain

Eq′8 = − Ψ(W )

(2 + r)2(1 + 2r)2
[
(g + 2m− 2gr + 14mr + gr2 + 2mr2

− (r − 1)(2 + r)(1 + 2r)W
]
,

Eq′10 =
Ψ(W )

(2 + r)3(1 + 2r)3
[
− c(2 + r)2(1 + 2r)2 + (g − 2m)(8g − 4m+ 11gr

+ 8mr + 8gr2 − 4mr2) + (g − 2m)(r − 1)(2 + r)(1 + 2r)W

− (2 + r)2(1 + 2r)2W 2
]
,

where Ψ(W ) = (g−2m)(r−1)+(2+r)(1+2r)W . So Eq′8 and Eq′10 have a common
factor which is linear in W . This means that the detected conditions are sufficient
for the existence of six invariant affine lines for system (3.44). As a result these
systems become

ẋ =
[
x+

3(g − 2m)

(2 + r)(1 + 2r)

][
c− 6(g − 2m)(g + gr + 3mr + gr2)

(2 + r)2(1 + 2r)2

+
2(g + gr + 3mr + gr2)

(2 + r)(1 + 2r)
x+ rx2

]
≡ L̃1(x)L̃2,3(x),

ẏ =
[
y − (g − 2m)(r − 1)

(2 + r)(1 + 2r)

][
c− (g − 2m)(8g − 4m+ 11gr + 8mr + 8gr2 − 4mr2)

(2 + r)2(1 + 2r)2

+
(g + 2m− 2gr + 14mr + gr2 + 2mr2)

(2 + r)(1 + 2r)
x+

(g − 2m)(r − 1)

(2 + r)(1 + 2r)
y

− (1− r)xy + y2
]
.

We need to detect if the two lines defined by the equation L̃2,3 = 0 are real or
complex and in the case when they are real, if one of them coincides with the
invariant line L̃1 = 0. So we calculate

Discrim[L̃2,3, x] =
λ̃(c, g,m, r)

(2 + r)2(1 + 2r)2
, Resx(L̃1, L̃2,3) =

µ̃(c, g,m, r)

(2 + r)2(1 + 2r)2
,

where

λ̃ = (g + 7gr − 9mr + gr2)(g + gr + 3mr + gr2)− cr(2 + r)2(1 + 2r)2,

µ̃ = c(2 + r)2(1 + 2r)2 − 3(g − 2m)(4g + gr + 18mr + 4gr2).
(3.98)

We observe that

sign
(

Discrim[L̃2,3, x]
)

= sign(λ̃),



58 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

i.e. the invariant lines L̃2,3 = 0 are real (respectively complex; coinciding) if λ̃ > 0

(respectively λ̃ < 0; λ̃ = 0). And the invariant line L̃1 = 0 coincides with one of

the lines L̃2,3 = 0 if and only if µ̃ = 0.

1.1.1. Case λ̃ > 0. Then we use a new parameter u setting λ̃ = u2 and since
r(2 + r)(1 + 2r) 6= 0 we obtain

c =
1

r(2 + r)2(1 + 2r)2
[
(g + 7gr − 9mr + gr2)(g + gr + 3mr + gr2)− u2

]
. (3.99)

This leads to the system

ẋ =r
[
x+

3(g − 2m)

(2 + r)(1 + 2r)

][
x+

g + gr + 3mr + gr2 − u
r(2 + r)(1 + 2r)

]
×
[
x+

g + gr + 3mr + gr2 + u

r(2 + r)(1 + 2r)

]
,

ẏ =
[
y − (g − 2m)(r − 1)

(2 + r)(1 + 2r)

][g2(r − 1)2(r + 1)2

r(r + 2)2(2r + 1)2
+

2gm(7r2 + 13r + 7)

(r + 2)2(2r + 1)2

− m2r(8r2 + 11r + 8) + u2

r(r + 2)2(2r + 1)2
+
g(r − 1)2 + 2m(1 + 7r + r2)

(2 + r)(1 + 2r)
x

+
(g − 2m)(r − 1)

(2 + r)(1 + 2r)
y + (r − 1)xy + y2

]
.

(3.100)

On the other hand for the value of c given in (3.99) we calculate

µ̃ =
1

r
(γ̃2 − u2), γ̃ = g(r − 1)2 + 9mr (3.101)

and since the condition µ̃ = 0 leads to the coincidence of two invariant lines of the
triplet, we examine two subcases: µ̃ 6= 0 and µ̃ = 0.

1.1.1.1. Subcase µ̃ 6= 0. Then (γ̃ − u)(γ̃ + u) 6= 0 and via the transformation

x1 = αx− 3(g − 2m)r

γ̃ − u
, y1 = αy +

(g − 2m)(r − 1)r

γ̃ − u
,

t1 =
(γ̃ − u)2

r2(2 + r)2(1 + 2r)2
t, α = −r(2 + r)(1 + 2r)

γ̃ − u
system (3.100) could be brought to the following family of systems (we keep the
old notations for the variables):

ẋ = rx(x− 1)(x− v), ẏ = y
[
rv − r(1 + v)x+ (r − 1)xy + y2

]
. (3.102)

It remains to note that this family of systems is a subfamily of (3.59) defined by
the condition p = 0 and hence it is already examined.

1.1.1.2. Subcase µ̃ = 0. Then (γ̃ − u)(γ̃ + u) = 0 and we may assume γ̃ + u = 0
because of the change u → −u. So setting u = −γ̃ 6= 0 in system (3.100) and
applying the transformation

x1 = αx− 3(g − 2m)r

2γ̃
, y1 = αy +

(g − 2m)(r − 1)r

2γ̃
,

t1 =
4γ̃2

p2(r − 1)2(p+ r)2
t, α = −r(2 + r)(1 + 2r)

2γ̃

we obtain the family of systems:

ẋ = rx2(x− 1), ẏ = y
[
− rx+ (r − 1)xy + y2

]
.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 59

We observe that this family of systems is a subfamily of (3.62) defined by the
condition p = 0, i.e. no new configurations could be obtained.

1.1.2. Case λ̃ < 0. In this case we set λ̃ = −u2 and since r(2 + r)(1 + 2r) 6= 0
we obtain

c =
1

r(2 + r)2(1 + 2r)2
[
(g+ 7gr− 9mr+ gr2)(g+ gr+ 3mr+ gr2) + u2

]
. (3.103)

This leads to the system

ẋ =r
[
x+

3(g − 2m)

(2 + r)(1 + 2r)

][(
x+

g + gr + 3mr + gr2

r(2 + r)(1 + 2r)

)2
+

u2

r2(2 + r)2(1 + 2r)2

]
,

ẏ =
[
y − (g − 2m)(r − 1)

(2 + r)(1 + 2r)

][g2(r − 1)2(r + 1)2

r(r + 2)2(2r + 1)2
+

2gm(7r2 + 13r + 7)

(r + 2)2(2r + 1)2

− m2r(8r2 + 11r + 8)− u2

r(r + 2)2(2r + 1)2
+
g(r − 1)2 + 2m(1 + 7r + r2)

(2 + r)(1 + 2r)
x

+
(g − 2m)(r − 1)

(2 + r)(1 + 2r)
y + (r − 1)xy + y2

]
(3.104)

and we consider two subcases: γ̃ 6= 0 and γ̃ = 0 (γ̃ is given in (3.101)).

1.1.2.1. Subcase γ̃ 6= 0. Then via the transformation

x1 = αx+
3(g − 2m)r

γ̃
, y1 = αy − (g − 2m)(r − 1)r

γ̃
,

t1 =
γ̃2

r2(2 + r)2(1 + 2r)2
t, α =

r(2 + r)(1 + 2r)

γ̃

the above system can be brought to the canonical form

ẋ = rx
[
(x+ 1)2 + v2

]
, ẏ = y

[
r(1 + v2) + 2rx+ (r − 1)xy + y2

]
, (3.105)

where v = u/γ̃ 6= 0. We observe that these systems belong to the family (3.64) for
p = 0 and this family is already examined.

1.1.2.2. Subcase γ̃ = 0. Considering (3.101) we obtain m = −g(r − 1)2/(9r)
and then system (3.104) becomes system

ẋ =r
[
x+

g)

3r

][(
x+

g)

3r

)2
+

u2

r2(2 + r)2(1 + 2r)2

]
,

ẏ =−
[
y − g(r − 1)

9r

][2g2(−1 + r)2(2 + r)2(1 + 2r)2 − 81ru2

81r2(2 + r)2(1 + 2r)2

+
g(r − 1)2

9r
x− g(r − 1)

9r
y + (1− r)xy − y2

]
.

Since ur(2 + r)(1 + 2r) 6= 0 we can apply the transformation

x1 = αx+
g(2 + r)(1 + 2r)

3u
, y1 = αy − g(r − 1)(2 + r)(1 + 2r)

9u
,

t1 =
u2

r2(2 + r)2(1 + 2r)2
t, α =

r(2 + r)(1 + 2r)

u



60 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

and we arrive at the family of systems

ẋ = rx(x2 + 1), ẏ = y
[
r + (r − 1)xy + y2

]
. (3.106)

It remains to observe that this family of systems is a subfamily of (3.68) defined
by p = 0, i.e. no new configurations could be obtained.

1.1.3. Case λ̃ = 0. Considering (3.98) and solving the equation λ̃ = 0 with
respect to the parameter c it is clear that we obtain (3.103) for u = 0. This leads
to system (3.104) with u = 0, which we denote by (3.104){u=0}.

We observe that in this case we obtain µ̃ = γ̃2/r and we again consider two
subcases: µ̃ 6= 0 and µ̃ = 0.

1.1.3.1. Subcase µ̃ 6= 0. Then γ̃ 6= 0 and via the transformation

x1 = αx− g + gr + 3mr + gr2

γ̃
, y1 = αy +

(g − 2m)(r − 1)r

γ̃
,

t1 =
γ̃2

r2(2 + r)2(1 + 2r)2
t, α =

r(2 + r)(1 + 2r)

γ̃

system (3.104){u=0} can be brought to the family of systems

ẋ = rx2(x− 1), ẏ = y
[
− r + 2rx+ (1− r)y + (r − 1)xy + y2

]
. (3.107)

We notice that the above systems belong to the family (3.69) for which p = 0 and
this family is already examined.

1.1.3.2. Subcase µ̃ = 0. In this case we have λ̃ = µ̃ = 0 and this leads to the
existence a triple invariant line in the direction x = 0. Since µ̃ = γ̃2/r we obtain
γ̃ = 0 and considering (3.101) we obtain m = −g(r − 1)2/(9r). Therefore taking
into account (3.103) for u = 0 we have c = g2/(3r) and then for these values of the
parameters m, u and c system (3.104) becomes

ẋ =
(g + 3rx)3

27r2
, ẏ =

(gr − g − 9ry)2)

729r3
[
2g(−1 + r) + 9(−1 + r)rx+ 9ry

]
.

However these systems possess invariant lines of total multiplicity 8, because for
these system we have (see Notation 2.4):

H(X,Y, Z) =
α

r11
(3rX + gZ)3(−9rY − gZ + grZ)2(9rX − 9rY + 2gZ + grZ)

× (9r2X + 9rY + gZ + 2grZ),

where α ∈ R. So we are out of the class of system studied in this article.

1.2. Possibility (1 + 2r)(2 + r) = 0. Considering Remark 3.9 we may assume
2r + 1 = 0 because of the rescaling (x, y, t) 7→ (x,−ry, t/r2) in system (3.44). So
r = −1/2 and then the equations (3.91) and (3.92) become, respectively:

Eq′5 = 0, Eq′8 = e− 2mW − 3W 2/2, Eq′10 = b− fW −W 3,

and

Eq′′6 = −g + 2m, Eq′′9 = −c+ e+ f − gW + 3W 2/2,

Eq′′10 = −a+ b+ (e− c)W − gW 2 +W 3/2.

So the condition Eq′′6 = 0 gives g = 2m and therefore equation (3.93) becomes

Eq′′′6 = 3W/2, Eq′′′9 = −c+ 2e+ f − 4mW + 3W 2,

Eq′′′10 = b− a/2 + (2e− c)W − 4mW 2 + 2W 3.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 61

We observe that the equation Eq′′′6 = 0 gives W = 0 and hence we obtain the
conditions:

Eq′′′9 = −c+ 2e+ f = 0, Eq′′′10 = b− a/2 = 0.

Thus we obtain f = c − 2e and a = 2b and then we calculate ResW (Eq′′9 , Eq
′′
10) =

−ResW (Eq′8, Eq′10) = Ψ̃(b, c, e,m)/8, where

Ψ̃(b, c, e,m) = 27b2 − 18c2e− 32e3 + 64e2m2 + 64bm3 + 4c(12e2 + 9bm− 8em2).

So in order to have invariant lines of total multiplicity seven the condition Ψ̃ = 0
is necessary. This equation is quadratic in b and we calculate

Discrim[Ψ̃, b] = 8(3e+ 2m2)(9c− 12e+ 16m2)2

and clearly we could have a real solution of the equation Ψ̃ = 0 with respect to the
parameter b only if either (3e + 2m2) ≥ 0 or 9c − 12e + 16m2 = 0. We consider
both cases.

1.2.1. Case (3e + 2m2) ≥ 0. Then setting 3e + 2m2 = 2u2 ≥ 0 we obtain
e = 2(u2 −m2) and we have

Ψ̃ = (27b+ 18cm+ 32m3 + 18cu+ 48m2u− 16u3)

× (27b+ 18cm+ 32m3 − 18cu− 48m2u+ 16u3)/27 = 0.

Because of the change u → −u we assume without loss of generality that the first
factor vanishes and we have the condition

b = −2(m+ u)(9c+ 16m2 + 8mu− 8u2)/27.

Then considering also the conditions

k = d = h = p = l = 0, r = −1/2, g = 2m, f = c− 2e, a = 2b (3.108)

we detect that system (3.44) has the form

ẋ = − (4m+ 4u− 3x)

54

[
2(9c+ 16m2 + 8mu− 8u2) + 12(2m− u)x− 9x2

]
)

≡ −L′1(x)L′2,3(x)/54,

ẏ = − (2m+ 2u− 3y)

54

[
2(9c+ 16m2 + 8mu− 8u2) + 18(m− u)x

+ 12(m+ u)y − 27xy + 18y2
]
.

(3.109)
Following the same algorithm as before we calculate

Discrim[L′2,3, x] = 216(3c+ 8m2 − 2u2) ≡ 216λ′(c,m, u),

Resx(L′1, L
′
2,3) = 54(3c+ 8m2 − 8u2) ≡ 54µ′(c,m, u)

(3.110)

and hence the invariant lines L′2,3 = 0 are real (respectively complex) if λ′ > 0
(respectively λ′ < 0) and they coincide if λ′ = 0. On the other hand the third line
(L′1 = 0) of this triplet coalesces with another invariant line if and only if µ′ = 0.
So we examine the possibilities given by these conditions.



62 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

1.2.1.1. Subcase λ′ > 0. Setting λ′ = 6w2 > 0 we obtain c = −2(4m2 − u2 −
3w2)/3 and then system (3.109) becomes

ẋ =(4m+ 4u− 3x)(4m− 2u− 6w − 3x)(4m− 2u+ 6w − 3x)/54,

ẏ =(2m+ 2u− 3y)
[
4(2m− u− 3w)(2m− u+ 3w)− 18(m− u)x

− 12(m+ u)y + 27xy − 18y2
]
/54.

(3.111)

For the above value of the parameter c we calculate µ′ = −6(u−w)(u+w) and we
consider two possibilities: µ′ 6= 0 and µ′ = 0.

1.2.1.1.1. Possibility µ′ 6= 0. Then applying the transformation

x1 =
x

2(w − u)
+

2(m+ u)

3(u− w)
, y1 =

y

2(w − u)
+

m+ u

3(u− w)
, t1 = 4(w − u)2t

we obtain the 1-parameter family of systems:

ẋ = x(x− 1)(v − x)/2, ẏ = y
[
− v + (1 + v)x− 3xy + 2y2

]
/2.

We observe that this family is a subfamily of (3.102) defined by r = −1/2 and
hence no new configurations could be obtained.

1.2.1.1.2. Possibility µ′ = 0. In this case we may assume w = u (due to the
change w → −w) and system (3.111) becomes

ẋ =(4m− 8u− 3x)(4m+ 4u− 3x)2/54,

ẏ =(2m+ 2u− 3y)
[
16(m− 2u)(m+ u)− 18(m− u)x− 12(m+ u)y

+ 27xy − 18y2
]
/54.

Since in this case we have λ′ = 6u2 > 0 (i.e. u 6= 0) applying the transformation

x1 = − x

4u
+
m+ u

3u
, y1 = − y

4u
+
m+ u

6u
, t1 = 16u2t

we obtain the system

ẋ = x2(1− x)/2, ẏ = y(x− 3xy + 2y2)/2.

It remains to observe that this system is contained in the family (3.62) for p = 0
and r = −1/2, so this family is already examined.

1.2.1.2. Subcase λ′ < 0. Setting λ′ = −6w2 < 0 we obtain c = −2(4m2 − u2 +
3w2)/3 and then system (3.109) becomes

ẋ =(4m+ 4u− 3x)
[
(4m− 2u− 3x)2 + 36w2

]
/54,

ẏ = (2m+ 2u− 3y)
[
4(4m2 − 4mu+ u2 + 9w2)− 18(m− u)x

− 12(m+ u)y + 27xy − 18y2
]
/54.

(3.112)

For the above value of the parameter c we calculate µ′ = −6(u2 + w2) 6= 0 since
λ′ = −6w2 < 0. So we consider two possibilities: u 6= 0 and u = 0.

1.2.1.2.1. Possibility u 6= 0. Then we apply the transformation

x1 =
x

2u
− 2(m+ u)

3u
, y1 =

y

2u
− m+ u

3u
, t1 = 4u2t

obtaining the systems

ẋ = −x
[
(x+ 1)2 + v2

]
/2, ẏ = y(−1− v2 − 2x− 3xy + 2y2)/2,

where v = u/w 6= 0. We observe that the above systems form a subfamily of (3.105)
defined by r = −1/2 and hence no new configurations could be obtained.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 63

1.2.1.2.2. Possibility u = 0. In this case doing the transformation

x1 =
x

2w
− 2m

3w
, y1 =

y

2w
− m

3w
, t1 = 4w2t

system (3.112) can be brought to the system

ẋ = −x(x2 + 1)/2, ẏ = y(−1− 3xy + 2y2)/2.

It remains to notice that this system is contained in the family (3.106) for r = −1/2,
i.e. no new configurations could be obtained.

1.2.1.3. Subcase λ′ = 0. Considering (3.110) this condition gives c = −2(2m−
u)(2m+ u)/3 and then system (3.109) becomes

ẋ =(4m+ 4u− 3x)(4m− 2u− 3x)2/54,

ẏ = (2m+ 2u− 3y)
[
4(2m− u)2 − 18(m− u)x− 12(m+ u)y

+ 27xy − 18y2
]
/54.

(3.113)

For the value of the parameter c, given above, we calculate µ′ = −6u2 and we
examine two possibilities: µ′ 6= 0 and µ′ = 0.

1.2.1.3.1. Possibility µ′ 6= 0. Then u 6= 0 and performing the transformation

x1 =
x

2u
− 2m− u

3u
, y1 =

y

2u
− m+ u

3u
, t1 = 4u2t

we obtain the system

ẋ = −x2(x− 1)/2, ẏ = y(1− 2x+ 3y − 3xy + 2y2)/2.

We notice that this system belongs to the family (3.107) for r = −1/2, already
examined.

1.2.1.3.2. Possibility µ′ = 0. Therefore u = 0 and systems (3.113) become

ẋ = (4m− 3x)3/54, ẏ = (2m− 3y)2(8m− 9x+ 6y)/54.

For these systems we calculate (see Notation 2.4)

H(X,Y, Z) = 2−43−9(X − 2Y )(3X − 4mZ)3(3X − 3Y − 2mZ)2(3Y − 2mZ)2

and hence by Lemma 2.6 we have invariant lines of total multiplicity nine, i.e. we
are not in the class of systems with invariant lines of total multiplicity exactly
seven.

1.2.2. Case 9c− 12e+ 16m2 = 0. Then we obtain c = 4(3e− 4m2)/9 and this

implies Ψ̃ = (9b+ 8em)2/3. Therefore the condition Ψ̃ = 0 yields b = −8em/9 and
considering also the conditions (3.108) we detect that system (3.44) becomes

ẋ =− (4m− 3x)(8e+ 8mx− 3x2)/18,

ẏ =− (8m− 9x+ 6y)(2e+ 4my − 3y2)/18.

However for these systems, calculations yield

H(X,Y, Z) = −2−43−7(X − 2Y )(3X − 4mZ)(3X2 − 8mXZ − 8eZ2)(3Y 2

− 4mY Z − 2eZ2)(3X2 − 6XY + 3Y 2 − 4mXZ + 4mY Z − 2eZ2)

and according to Lemma 2.6 the above systems have invariant lines of total multi-
plicity nine, so we are out of the class we examine in this work.

2. Subcase p 6= 0 and (1+p+2r)(p+2r+r2) = 0. We claim that in this case we
may assume 1 + p+ 2r = 0 because the case p+ 2r+ r2 = 0 can be reduced to the



64 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

first one via a rescaling. Indeed, considering Remark 3.9 we observe that by the
rescaling (x, y, t) 7→ (x,−ry, t/r2) in system (3.44) (which replaces the direction
y = −rx with y = x) the cubic homogeneous system associated to these system
becomes

ẋ1 = (p1 + r1)x31, ẏ1 = p1x
2y1 + (r1 − 1)x1y

2
1 + y31 ,

where p1 = p/r2 and r1 = 1/r. Therefore,

p1 + 2r1 + r21 =
p

r2
+

2

r
+

1

r2
=

1

r2
(p+ 2r + 1) = 0

and this completes the proof of our claim.
So p = −(2r + 1) 6= 0 and considering the condition k = d = h = 0 (for

the existence of a triplet in the direction x = 0) and (3.46) we detect that the
condition Eq′′6 = 0 gives l = g − 2m. Now taking into consideration the conditions
k = d = h = 0, p = −(2r + 1), l = g − 2m as well as the equations (3.45), (3.46)
and (3.47) for the remaining three directions, we calculate:

(a) for the direction y = 0:

Eq′5 = g − 2m+ (1 + 2r)W, Eq′8 = e− 2mW + (r − 1)W 2,

Eq′10 = b− fW −W 3;
(3.114)

(b) for the direction y = x:

Eq′′6 = 0, Eq′′9 = −c+ e+ f − 2mW + (2 + r)W 2,

Eq′′10 = −a+ b+ (e− c)W − 2mW 2 + (1 + r)W 3;
(3.115)

(c) for the direction y = −rx:

Eq′′′6 =
r + 1

r

[
2m− g + (r − 1)W

]
,

Eq′′′9 = f − c− e/r +
g − 2m+ gr

r2
W +

1 + r + r2

r2
W 2,

Eq′′′10 = b+ ar − e+ cr

r
W +

g − 2m+ gr

r2
W 2 +

1 + r

r2
W 3.

(3.116)

Since 1 + 2r 6= 0 the equation Eq′5 = 0 is linear in W and hence we obtain W =
(2m− g)/(1 + 2r). Therefore we obtain the equations

Eq′8 =
e(1 + 2r)2 + (g − 2m)(−g + 4m+ gr + 2mr)

(1 + 2r)2
= 0,

Eq′10 =
b(1 + 2r)3 + (g − 2m)

[
(g − 2m)2 + f(1 + 2r)2

]
(1 + 2r)3

= 0,

which lead to the two conditions:

e = − (g − 2m)(4m− g + gr + 2mr)

(1 + 2r)2
, b = −

(g − 2m)
[
(g − 2m)2 + f(1 + 2r)2

]
(1 + 2r)3

.

We observe that the equation Eq′′′6 = 0 from (3.116) is linear in W if and only if
r− 1 6= 0. So in what follows we consider two possibilities: r− 1 6= 0 and r− 1 = 0.

2.1. Possibility r−1 6= 0. Then the equation Eq′′′6 = 0 gives W = (g−2m)/(r−1)
and considering the above conditions the equations Eq′′′9 = 0 and Eq′′′10 = 0 give us



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 65

the following two new conditions:

a =
3(g − 2m)

[
f(r − 1)2(1 + 2r)2 + 3(g − 2m)2(1 + r + r2)

]
(r − 1)3(1 + 2r)3

,

c =
f(r − 1)2(1 + 2r)2 + 3(g − 2m)

[
− 2m(2 + r)2 + 3g(1 + r + r2)

]
(r − 1)2(1 + 2r)2

.

So considering all the above conditions we could convince ourselves that
ResW (Eq′′9 , Eq

′′
10) = 0 which means that the obtained conditions guarantee the

existence of at least one invariant line in the direction y = 0.
Thus setting all these conditions in system (3.44) we arrive at the following

family of systems:

ẋ =
[
x+

3(g − 2m)

(r − 1)(1 + 2r)

][
f +

3(g − 2m)2(1 + r + r2)

(r − 1)2(1 + 2r)2

+
2(g − 3m+ gr − 3mr + gr2)

(r − 1)(1 + 2r)
x− (r + 1)x2

]
≡ L′1(x)L′2,3(x),

ẏ =
[
y − (g − 2m)

(1 + 2r)

][
f +

(g − 2m)2

(1 + 2r)2
+

4m− g + gr + 2mr

(1 + 2r)
x+

(g − 2m)

(1 + 2r)
y

− (1 + 2r)x2 + (r − 1)xy + y2
]
.

We again need to detect if the two lines defined by the equation L′2,3 = 0 are real
or complex and in the case when they are real, if one of them coincides with the
invariant line L′1 = 0. So we calculate

Discrim[L′2,3, x] = 4(r − 1)2(1 + 2r)2λ1(f, g,m, r),

Resx(L′1, L
′
2,3) = (r − 1)2(1 + 2r)2µ(1)(c, g,m, r)

where

λ1 = f(r − 1)2(1 + r)(1 + 2r)2 − 18gm(1 + r)(1 + r + r2)

+ g2(2 + r)2(1 + r + r2) + 3m2(1 + r)(7 + 7r + 4r2),

µ(1) = c(2 + r)2(1 + 2r)2 − 3(g − 2m)(4g + gr + 18mr + 4gr2).

(3.117)

We observe that

sign
(

Discrim[L′2,3, x]
)

= sign(λ1),

i.e. the invariant lines L′2,3 = 0 are real (respectively complex; coinciding) if λ1 > 0
(respectively λ1 < 0; λ1 = 0). The invariant line L′1 = 0 coincides with one of the
lines L′2,3 = 0 if and only if µ(1) = 0.

2.1.1. Case λ1 > 0. Then we may use a new parameter u setting λ1 = u2 and
since (r − 1)(1 + r)(1 + 2r) 6= 0 we obtain

f =
1

(r − 1)2(1 + r)(1 + 2r)2
[
18gm(1 + r)(1 + r + r2)

− g2(2 + r)2(1 + r + r2)− 3m2(1 + r)(7 + 7r + 4r2) + u2
]
.

(3.118)



66 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

This leads to the family of systems

ẋ = −(r + 1)
[
x+

3(g − 2m)

(r − 1)(1 + 2r)

][
x− g − 3m+ gr − 3mr + gr2 − u

(r − 1)(1 + r)(1 + 2r)

]
×
[
x− g − 3m+ gr − 3mr + gr2 + u

(r − 1)(1 + r)(1 + 2r)

]
,

ẏ = −
[
y − (g − 2m)

1 + 2r

][g2(r4 + 4r3 + 10r2 + 9r + 3)

(r − 1)2(r + 1)(2r + 1)2

− 2gm(7r2 + 13r + 7)

(r − 1)2(2r + 1)2
+
m2(r + 1)(8r2 + 29r + 17)− u2

(r − 1)2(r + 1)(2r + 1)2
− y2

+
4m− g + gr + 2mr

1 + 2r
x− (g − 2m)

1 + 2r
y + (1 + 2r)x2 + (1− r)xy

]
.

(3.119)

On the other hand for the value of f given in (3.118) we calculate

µ(1) = − 1

r + 1
(γ21 − u2), γ1 = g(2 + r)2 − 9m(r + 1) (3.120)

and since the condition µ(1) = 0 leads to the coalescence of two invariant lines of
the triplet, we examine two possibilities: µ(1) 6= 0 and µ(1) = 0.

2.1.1.1. Subcase µ(1) 6= 0. Then (γ1−u)(γ1 +u) 6= 0 and via the transformation

x1 = αx− 3(g − 2m)(1 + r)

γ1 + u
, y1 = αy − (g − 2m)(r − 1)(1 + r)

γ1 + u
,

t1 =
(γ1 + u)2

(r − 1)2(1 + r)2(1 + 2r)2
t, α =

(r − 1)(1 + r)(1 + 2r)

γ1 + u

system (3.100) can be brought to the following family of systems (we keep the old
notations for the variables):

ẋ =− (r + 1)x(x− 1)(x− v),

ẏ =y
[
− (1 + r)v + (1 + r)(1 + v)x− (1 + 2r)x2 + (r − 1)xy + y2

]
,

(3.121)

where v = (γ1 − u)/(γ1 + u) 6= 0. It remains to observe that this family of systems
is a subfamily of (3.59) defined by the condition p = −(2r + 1) which was already
examined.

2.1.1.2. Subcase µ(1) = 0. Then (γ1 − u)(γ1 + u) = 0 and we may assume
γ1 + u = 0 due to the change u → −u. So setting u = −γ1 6= 0 in system (3.119)
and applying the transformation

x1 = αx+
3(g − 2m)(1 + r)

2γ1
, y1 = αy − (g − 2m)(r − 1)(1 + r)

2γ1
,

t1 =
4γ21

(r − 1)2(1 + r)2(1 + 2r)2
t, α =

(r − 1)(1 + r)(1 + 2r)

2γ1

we obtain the family of systems

ẋ = −(r + 1)x2(x− 1), ẏ = y
[
(1 + r)x− (1 + 2r)x2 + (r − 1)xy + y2

]
. (3.122)

We observe that this family of systems is a subfamily of (3.62) defined by the
condition p = −(2r + 1), i.e. no new configurations could be obtained.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 67

2.1.2. Case λ1 < 0. Then we set λ1 = −u2 and since r(2 + r)(1 + 2r) 6= 0 we
obtain

f =
1

(r − 1)2(1 + r)(1 + 2r)2
[
18gm(1 + r)(1 + r + r2)

− g2(2 + r)2(1 + r + r2)− 3m2(1 + r)(7 + 7r + 4r2)− u2
]
.

(3.123)

This leads to the system

ẋ =− (r + 1)
[
x+

3(g − 2m)

(r − 1)(1 + 2r)

][(
x− g − 3m+ gr − 3mr + gr2

(r − 1)(1 + r)(1 + 2r)

)2
+

u2

(r − 1)2(1 + r)2(1 + 2r)2

]
,

ẏ =−
[
y − (g − 2m)

1 + 2r

][g2(r4 + 4r3 + 10r2 + 9r + 3)

(r − 1)2(r + 1)(2r + 1)2

− 2gm(7r2 + 13r + 7)

(r − 1)2(2r + 1)2
+
m2(r + 1)(8r2 + 29r + 17)− u2

(r − 1)2(r + 1)(2r + 1)2
− y2

+
4m− g + gr + 2mr

1 + 2r
x− (g − 2m)

1 + 2r
y + (1 + 2r)x2 + (1− r)xy

]
(3.124)

and we examine two subcases: γ1 6= 0 and γ1 = 0 (γ1 is given in (3.120)).

2.1.2.1. Subcase γ1 6= 0. Then via the transformation

x1 = αx− 3(g − 2m)(1 + r)

γ1
, y1 = αy +

(g − 2m)(r − 1)(1 + r)

γ1
,

t1 =
(γ1)2

(r − 1)2(1 + r)2(1 + 2r)2
t, α = − (r − 1)(1 + r)(1 + 2r)

γ1

the above system can be brought to the canonical form

ẋ =− (r + 1)x
[
(x+ 1)2 + v2

]
,

ẏ =y
[
− (1 + r)(1 + v2)− 2(1 + r)x− (1 + 2r)x2 + (r − 1)xy + y2

]
,

(3.125)

where v = u/γ1 6= 0. We observe that these systems belong to the family (3.64) for
which p = −(2r + 1) and hence no new configurations could be obtained.

2.1.2.2. Subcase γ1 = 0. Considering (3.120) we obtain m = g(2+r)2/(9(1+r))
and then systems (3.124) become

ẋ =− (1 + r)
[
x− g)

3(1 + r)

][(
x− g)

3(1 + r)

)2
+

u2

(−1 + r)2(1 + r)2(1 + 2r)2

]
,

ẏ =−
[
y +

g(r − 1)

9(1 + r)

][g2(r − 1)2(1 + 2r)2(11 + 14r + 2r2) + 81(1 + r)u2

81(r − 1)2(1 + r)2(1 + 2r)2

− g(7 + 10r + r2)

9(1 + r)
x+

g(r − 1)

9(1 + r)
y + (1 + 2r)x2 + (1− r)xy − y2

]
.

Since ur(1 + r)(r − 1)(1 + 2r) 6= 0 we can apply the transformation

x1 = αx− g(r − 1)(1 + 2r)

3u
, y1 = αy +

g(r − 1)2(1 + 2r)

9u
,

t1 =
u2

(r − 1)2(1 + r)2(1 + 2r)2
t, α =

(r − 1)(1 + r)(1 + 2r)

u



68 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

and we arrive at the family of systems

ẋ = −(1 + r)x(x2 + 1), ẏ = y
[
− 1− r − (1 + 2r)x2 + (r − 1)xy + y2

]
. (3.126)

It remains to observe that this is a subfamily of (3.68) defined by p = −(2r + 1).
So no new configurations could be obtained.

2.1.3. Case λ1 = 0. Considering (3.117) and solving the equation λ1 = 0 with
respect to the parameter f it is clear that we obtain (3.123) for u = 0. This leads
to system (3.124) with u = 0, which we denote by (3.124){u=0}. We observe that

in this case we obtain µ(1) = −γ21/(1 + r) and we again consider two subcases:
µ(1) 6= 0 and µ(1) = 0.

2.1.3.1. Subcase µ(1) 6= 0. Then γ1 6= 0 and via the transformation

x1 = αx+
g − 3m+ gr − 3mr + gr2

γ1
, y1 = αy +

(g − 2m)(r − 1)(1 + r)

γ1
,

t1 =
γ21

(r − 1)2(1 + r)2(1 + 2r)2
t, α = − (r − 1)(1 + r)(1 + 2r))

γ1

systems (3.124) with {u = 0} can be brought to the family of systems

ẋ =− (1 + r)x2(x− 1),

ẏ =y
[
− r + 2rx− (1 + 2r)x2 + (1− r)y + (r − 1)xy + y2

]
.

(3.127)

We notice that the above system belongs to the family (3.69) for p = −(2r+1) and
this family was already examined.

2.1.3.2. Possibility µ(1) = 0. In this case we have λ1 = γ1 = 0 and this leads to
the existence of a triple invariant line in the direction x = 0. Considering (3.120)
the condition γ1 = 0 gives m = g(2 + r)2/(9(1 + r)). Therefore taking into account
the condition u = 0 system (3.124) becomes

ẋ =− (r + 1)
[
x− g

3(1 + r)

]3
,

ẏ =
[
y +

g(r − 1)

9(1 + r)

][
− g2(11 + 14r + 2r2)

81(1 + r)2
+
g(7 + 10r + r2)

9(1 + r)
x− g(r − 1)

9(1 + r)
y

− (1 + 2r)x2 + (r − 1)xy + y2
]
.

However for these system we have (see Notation 2.4):

H(X,Y, Z) =
α

(1 + r)11
(3X + 3rX − gZ)3(9rX + 9r2X + 9Y + 9rY − gZ − 2grZ)

× (9X + 9rX − 9Y − 9rY − 2gZ − grZ)2(9Y + 9rY − gZ + grZ),

where α ∈ R. Therefore by Lemma 2.6 the above system have invariant lines of
total multiplicity 8, i.e. we are out of the family we study here.

2.2. Possibility r − 1 = 0. So r = 1 and considering (3.116) we obtain Eq′′′6 =
(r + 1)(2m − g)/r = 0 which implies g = 2m. Then from (3.114) we obtain that
W = 0 must be a common solution of the equations Eq′5 = Eq′8 = Eq′10 = 0. This
implies e = b = 0 and we obtain the following conditions on the parameters

k = d = h = l = e = b = 0, p = −3, g = 2m, r = 1, (3.128)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 69

which guarantees the existence of a triplet of parallel invariant lines in the direction
x = 0 and one invariant line in the direction y = 0. For the other two directions
y = x and y = −rx considering (3.115) and (3.116) we obtain respectively:

Eq′′9 = −c+ f − 2mW + 3W 2, Eq′′10 = −a− cW − 2mW 2 + 2W 3

and

Eq′′′9 = f − c+ 2mW + 3W 2, Eq′′′10 = a− cW + 2mW 2 + 2W 3.

We calculate ResW (Eq′′9 , Eq
′′
10) = ResW (Eq′′′9 , Eq

′′′
10) = Ψ′(a, c, f,m), where

Ψ′(a, c, f,m) = 27a2 + 2am(9c+ 4m2)− (c− f)(c2 + 4cf + 4f2 + 4fm2). (3.129)

So in order to have invariant lines of total multiplicity seven the condition Ψ′ = 0
is necessary. This equation is quadratic in a and we calculate

Discrim[Ψ′, a] = 4(3c− 3f +m2)(3c+ 6f + 4m2)2

and clearly we could have a real solution of the equation Ψ′ = 0 with respect to the
parameter a only if either (3c− 3f +m2) ≥ 0 or (3c+ 6f + 4m2) = 0. We consider
each one of these cases.

2.2.1. Case (3c− 3f +m2) ≥ 0. Then setting 3c− 3f +m2 = u2 ≥ 0 we obtain
f = (3c+m2 − u2)/3 and we obtain

Ψ′ = (27a+ 9cm+ 4m3 + 9cu+ 6m2u− 2u3)

×
(
27a+ 9cm+ 4m3 − 9cu− 6m2u+ 2u3

)
/27 = 0.

From the change u → −u we may assume without loss of generality that the first
factor vanishes and we have the condition

a = −(m+ u)(9c+ 4m2 + 2mu− 2u2)/27.

So considering also the conditions (3.128) we detect that system (3.44) has the form

ẋ = − (m+ u− 3x)

27

[
9c+ 4m2 + 2mu− 2u2 + 6(2m− u)x− 18x2

]
)

≡ −L′′1(x)L′′2,3(x)/27,

ẏ = y(3c+m2 − u2 + 6mx− 9x2 + 3y2)/3.

(3.130)

So we need again to detect if the two lines defined by the equation L′′2,3 = 0 are
real or complex and in the case when they are real, if one of them coincides with
the invariant line L′′1 = 0. So we calculate

Discrim[L′′2,3, x] = 108(6c+ 4m2 − u2) ≡ 108λ2(c,m, u),

Resx(L′′1 , L
′′
2,3) = 27(3c+ 2m2 − 2u2) ≡ 27µ(2)(c,m, u).

(3.131)

Therefore sign
(

Discrim[L′′2,3, x]
)

= sign(λ2) and hence the invariant lines L′′2,3 = 0
are real (respectively complex; coinciding) if λ2 > 0 (respectively λ2 < 0; λ2 = 0).
The invariant line L′′1 = 0 coincides with one of the lines L′′2,3 = 0 if and only if

µ(2) = 0.

2.2.1.1. Subcase λ2 > 0. Then we may use a new parameter w setting λ1 = 3w2

and we obtain c = (u2 + 3w2 − 4m2)/6. This leads to the system

ẋ =(2m− u− 3w − 6x)(2m− u+ 3w − 6x)(m+ u− 3x)/54,

ẏ =− y(2m2 + u2 − 3w2 − 12mx+ 18x2 − 6y2)/6.
(3.132)



70 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

On the other hand for the value of c given above, we calculate µ(2) = −3(u −
w)(u+w)/2 and since the condition µ(2) = 0 leads to the coalescence of two invariant
lines of the triplet, we examine two possibilities: µ(2) 6= 0 and µ(2) = 0.

2.2.1.1.1. Possibility µ(2) 6= 0. Then (u− w)(u + w) 6= 0 and via the transfor-
mation

x1 =
2x

w − u
+

2(m+ u)

3(u− w)
, y1 =

2y

w − u
, t1 = (u− w)2t/4

system (3.132) can be brought to the following family of systems (we keep the old
notations for the variables):

ẋ = −2x(x− 1)(x− v), ẏ = y
[
− 2v + 2(1 + v)x− 3x2 + y2

]
,

where v = (u+w)/(u−w) 6= 0. It remains to observe that this family of systems is
a subfamily of (3.121) defined by the condition r = 1 and this family was already
examined.

2.2.1.1.2. Possibility µ(2) = 0. In this case (u − w)(u + w) = 0 and we may
assume w = u (because of the change w → −w). Then system (3.132) becomes

ẋ =2(m− 2u− 3x)(m+ u− 3x)2/27, ẏ = −y(m2 − u2 − 6mx+ 9x2 − 3y2)/3.

Since in this case we have λ2 = 3u2 > 0 (i.e. u 6= 0) via the transformation

x1 = −x
u

+
m+ u

3u
, y1 = −y

u
, t1 = u2t

we obtain the system

ẋ = −2x2(1− x), ẏ = y(2x− 3x2 + y2).

We notice that this system is contained in the family (3.122) for r = 1, i.e. no new
configurations could be obtained.

2.2.1.2. Subcase λ2 < 0. Setting λ2 = −3w2 < 0 we obtain c = (u2 − 3w2 −
4m2)/6 and then system (3.130) becomes

ẋ =(m+ u− 3x)
[
(2m− u− 6x)2 + 9w2

]
/54,

ẏ =− y(2m2 + u2 + 3w2 − 12mx+ 18x2 − 6y2)/6.
(3.133)

For the above value of the parameter c we calculate µ(2) = −3(u2 +w2)/2 6= 0 since
λ2 = −3w2 < 0. We consider two possibilities: u 6= 0 and u = 0.

2.2.1.2.1. Possibility u 6= 0. Then we may apply the transformation

x1 =
2x

u
− 2(m+ u)

2u
, y1 =

2y

u
, t1 = u2t/4

getting the system

ẋ = −2x
[
(x+ 1)2 + v2

]
, ẏ = −y(2 + 2v2 + 4x+ 3x2 − y2),

where v = w/u 6= 0. We observe that the above systems form a subfamily of the
family (3.125) defined by r = 1 and this family was already examined.

2.2.1.2.2. Possibility u = 0. In this case via the transformation

x1 =
x

2w
− 2m

3w
, y1 =

y

2w
− m

3w
, t1 = 4w2t

system (3.133) will be brought to the system

ẋ = −2x(x2 + 1), ẏ = y(−2− 3x2 + y2).



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 71

It remains to notice that this system is contained in the family (3.126) for r = 1,
i.e. no new configurations could be obtained.

2.2.1.3. Subcase λ2 = 0. Considering (3.131) this condition gives c = (u2 −
4m2)/6 and then system (3.130) becomes

ẋ =(2m− u− 6x)2(m+ u− 3x)/54,

ẏ =− y(2m2 + u2 − 12mx+ 18x2 − 6y2)/6.
(3.134)

For the above value of the parameter c we calculate µ(2) = −3u2/2 and we examine
two possibilities: µ(2) 6= 0 and µ(2) = 0.

2.2.1.3.1. Possibility µ(2) 6= 0. Then u 6= 0 and via the transformation

x1 =
2x

u
− 2m− u

3u
, y1 =

2y

u
, t1 = u2t/4

we obtain the system

ẋ = −2(−1 + x)x2, ẏ = y(−1 + 2x− 3x2 + y2).

We notice that this system belongs to the family (3.127) for r = 1, i.e. no new
configurations could be obtained.

2.2.1.3.2. Possibility µ(2) = 0. In this case u = 0 and system (3.134) becomes

ẋ = 2(m− 3x)3/29, ẏ = −y(m2 − 6mx+ 9x2 − 3y2)/3.

For these systems we calculate (see Notation 2.4)

H(X,Y, Z) = 23−9Y (3X −mZ)3(3X + 3Y −mZ)2(−3X + 3Y +mZ)2

and hence, by Lemma 2.6 we have invariant lines of total multiplicity nine, i.e.
we are not in the class of systems with invariant lines of total multiplicity exactly
seven.

2.2.2. Case (3c + 6f + 4m2) = 0. Then we obtain c = −2(3f + 2m2)/3 and
this implies (see (3.129)) Ψ′ = (27a − 18fm − 8m3)2/27. Therefore the condition
Ψ′ = 0 yields a = 2(9fm+ 4m3)/27 and considering also the conditions (3.128) we
detect that systems (3.44) have the form

ẋ = 2(m− 3x)(9f + 4m2 − 6mx+ 9x2)/27, ẏ = y(f + 2mx− 3x2 + y2).

However for these systems calculations yield

H(X,Y, Z) = −2 · 3−8Y (3X −mZ)(9X2 − 6mXZ + 9fZ2 + 4m2Z2)

× (9X2 + 18XY + 9Y 2 − 6mXZ − 6mY Z + 9fZ2 + 4m2Z2)

× (9X2 − 18XY + 9Y 2 − 6mXZ + 6mY Z + 9fZ2 + 4m2Z2)

and according to Lemma 2.6 the above system has invariant lines of total multi-
plicity nine.

Existence of a triplet in a direction different from x = 0. Considering Remark 3.9
it is sufficient to consider two cases: when either the triplet exists in the direction
y = 0, or y = x.

We claim that in each one of these cases the corresponding systems (3.44) could
be brought via an affine transformation to systems, which form a subfamily of
systems (3.44) possessing a triplet in the direction x = 0. To prove this claim we
consider each one of the possibilities.



72 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

1. A triplet in the direction y = 0. Considering (3.45) it is clear that for the
existence of a triplet in this direction it is necessary and sufficient l = p = e = m = 0
and r = 1. In this case system (3.44) becomes

ẋ = a+ cx+ dy + gx2 + 2hxy + ky2 + x3, ẏ = b+ fy + y3.

So by the transformation x1 = y, y1 = x this system becomes

ẋ1 = a1 + c1x1 + x31, ẏ1 = b1 + e1x1 + f1y1 + l1x
2
1 + 2m1x1y1 + n1y

2
1 + y31

where

a1 = b, b1 = a, c1 = f, e1 = d, f1 = c, l1 = k, m1 = h, n1 = g

are free parameters. Then we may consider n1 = 0 due to a translation and going
back to the old notations we obtain the family of systems

ẋ = a+ cx+ x3, ẏ = b+ ex+ fy + lx2 + 2mxy + y3

which possess a triplet of invariant lines in the direction x = 0. Evidently this family
of systems is a subfamily of (3.44) defined by the conditions p = d = h = k = g = 0
and r = 1 and hence our claim is proved in this case.

2. Triplet in the direction y = x. In this case considering (3.46) we must impose
the equations Eq′′6 = 0 and Eq′′9 = 0 to vanish identically. So by (3.46) we arrive
at the conditions

c = −d+ e+ f, k = −h+m, l = g + h−m, p = 3, r = −2

and this leads to the family of systems

ẋ = a+ (e− d+ f)x+ dy + gx2 + 2hxy + (m− h)y2x3,

ẏ = b+ ex+ fy + (g + h−m)x2 + 2mxy + 3x2y − 3xy2 + y3.

Applying the transformation x2 = x−y and y2 = −y these systems can be brought
to the form

ẋ2 = a2 + c2x2 + g2x
2
2 + x32,

ẏ2 = b2 + e2x2 + f2y2 + l2x
2
2 + 2m2x2y2 + n2y

2
2 − 3xy2 + y3,

where

a2 = a− b, b2 = −b, c2 = f − d, e2 = −e, f2 = e+ f, g2 = m− h,
l2 = m− g − h, m2 = g + h, n2 = −(g + h+m)

are free parameters. We may consider n2 = 0 via a translation and returning to
the old notations we obtain the family of systems

ẋ = a+ cx+ gx2 + x3, ẏ = b+ ex+ fy + lx2 + 2mxy + ny2 − 3xy2 + y3

which evidently possess a triplet of invariant lines in the direction x = 0. It re-
mains to observe that this family of systems is a subfamily of (3.44) defined by the
conditions p = d = h = k = 0, p = 3 and r = −2 and this completes the proof of
our claim.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 73

3.4. Systems with configuration type T = (2, 2, 2). In this subsection we con-
struct a cubic system with four real infinite singular points which has six invariant
affine straight lines with the configuration of the type T = (2, 2, 2), having to-
tal multiplicity seven, as always the invariant straight line at infinity is included.
According to Theorem 2.10, in this case the condition V3 = 0 is necessary.

In [6, Subsection 3.4.1] it was proved that in this case via a linear transformation
and time rescaling the associated cubic homogeneous system can be brought to the
form

ẋ = rx3 + (2 + r)x2y, ẏ = (1 + 2r)xy2 + y3. (3.135)

Consider generic cubic systems with cubic homogeneities (3.135). Since r 6= 0 via
a translation we may assume n = 0 in system (2.4), i.e. a system possessing invariant
lines in the configuration (2, 2, 2) necessarily belongs to the following family:

ẋ = a+ cx+ dy + gx2 + 2hxy + ky2 + rx3 + (2 + r)x2y,

ẏ = b+ ex+ fy + lx2 + 2mxy + (1 + 2r)xy2 + y3, r(r + 1) 6= 0.
(3.136)

In what follows we shall determine necessary and sufficient conditions for a sys-
tems (3.136) to have invariant affine straight lines with the configuration of the
type T = (2, 2, 2).

Considering Remark 2.13 for the homogeneous systems (3.135), corresponding
to system (3.136), we calculate

H(X,Y, Z) = gcd(G1,G2,G3) = X2(X − Y )Y 2(rX + Y )2.

So each one of the invariant lines x = 0, y = 0 and rx+y = 0 of system (3.135) is of
multiplicity two and the line y = x is of multiplicity one. However for some values
of the parameter r the common divisor gcd(G1,G2,G3) could contain additional
factors (see Notation 2.4 and Lemma 2.6). Since the factor (X − Y ) depends on X
as well as on Y , in order to increase its multiplicity it is necessary

ResX(G2/H, G1/H) = ResY (G2/H,G1/H) = 0.

We calculate:

ResX(G2/H,G1/H) = 24(r − 1)(1 + r)2(2 + r)(1 + 2r)Y 3 = 0,

ResY (G2/H,G1/H) = −24(r − 1)(1 + r)2(2 + r)(1 + 2r)X3 = 0,

and since r(r+ 1) 6= 0 the condition (r− 1)(2 + r)(1 + 2r) = 0 must hold. However
for systems (3.135) with the conditions (r− 1)(2 + r)(1 + 2r) = 0 calculations yield

H(X,Y, Z) =


3X2(X − Y )Y 2(X + Y )3 if r = 1;

−6X3(X − Y )(2X − Y )2Y 2 if r = −2;

3X2(X − 2Y )2(X − Y )Y 3/8 if r = −1/2,

and hence, by Remark 2.13 we deduce that systems (3.136) could not possess a
couple of parallel invariant line in the direction y = x. So systems (3.136) could
possess three couples of invariant straight lines only in the directions x = 0, y = 0
and rx + y = 0. In [6, Subsection 3.4.2] these directions were examined and the
following statement was proved:



74 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Lemma 3.16 ([6]). Systems (3.136) have one couple of parallel invariant lines in
the direction x = 0 and one such couple in the direction y = 0 if and only if the
following conditions hold:

(r + 2)(2r + 1) 6= 0, k = l = h = 0, g = 2m, d = c(2 + r)/r,

a = 2cm/r, e = f(1 + 2r) + 4m2/(1 + 2r),

b = −2
[
4m3 + fm(1 + 2r)2

]
/(1 + 2r)3.

(3.137)

We examine the third direction y = −rx in which we could also have a couple of
parallel invariant straight lines. Considering equation (2.5), Remark 2.12 and the
conditions (3.137) we obtain

Eq8 = (1 + r)(f − cr) +
4m2

1 + 2r
− 2mW − (r − 1)W 2 = 0,

Eq10 = 2cm− 8m3

(1 + 2r)3
− 2fm

1 + 2r
− (2c+ f + cr)W −W 3 = 0,

R
(1)
W (Eq8, Eq10) = 2c(r − 1)− 2fr(r − 1)− 12m2r

1 + 2r
= 0.

It is clear that in order to have two common solutions of the equations Eq8 = 0
and Eq10 = 0 the condition r − 1 6= 0 is necessary. Therefore we obtain c =

fr + 6m2r
(r−1)(1+2r) and we calculate

R
(0)
W (Eq8, Eq10) = −

144m2r2(1 + r)2
[
f(r − 1)2(1 + 2r)2 + 3m2(1− 2r + 4r2)

]2
(r − 1)3(1 + 2r)6

= 0

and we have either m = 0 or m 6= 0 and

f = − 3m2(1− 2r + 4r2)

(−1 + r)2(1 + 2r)2
.

We claim, that in the case m = 0 we arrive at systems possessing invariant lines
of total multiplicity 8. Indeed, suppose m = 0. Then considering (3.137) and the
above expression for the parameter c we obtain the conditions

k = l = h = g = m = a = b = 0, c = fr, d = f(2 + r), e = f(1 + 2r).

Thus we obtain the family of systems

ẋ = (f + x2)(rx+ 2y + ry), ẏ = (f + y2)(x+ 2rx+ y)

with the condition r(r2−1)(r+ 2)(2r+ 1) 6= 0. This system possesses the following
seven invariant affine straight lines:

x2 + f = 0, y2 + f = 0, y − x = 0, (rx+ y)2 + f(1 + r)2 = 0

and this proves our claim.
In what follows we assume m 6= 0 and we examine the second possibility: f =

− 3m2(1−2r+4r2)
(−1+r)2(1+2r)2 . Then we obtain c = − 9m2r

(−1+r)2(1+2r)2 and taking into account the

conditions (3.137) this leads to the 2-parameter family of systems:

ẋ =
[
x− 3m

(r − 1)(1 + 2r)

][
x+

3m

(r − 1)(1 + 2r)

][
rx+ (2 + r)y + 2m

]
,

ẏ =
[
y − m

(r − 1)

][
y +

m(4r − 1)

(r − 1)(1 + 2r)

][
(1 + 2r)x+ y − 2m

1 + 2r

]
.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 75

As it was mentioned earlier, for these systems the condition mr(r+ 1)(r+ 2)(2r+
1)(r − 1) 6= 0 must hold. Then via the transformation

(x, y, t) 7→
( 3m(1− 2x)

(r − 1)(1 + 2r)
,
m(1 + 2r + 6ry)

(r − 1)(1 + 2r)
,

(r − 1)2(1 + 2r)2

36m2r
t
)

we arrive at the 1-parameter family of systems

ẋ = x(x− 1)
[
x− (r+ 2)y− 1− r

]
, ẏ = y(y+ 1)

[
ry− (1 + 2r)x+ 1 + r

]
. (3.138)

These systems possess six distinct invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : y = 0, L4 : y = −1, L5 : y = x, L6 : y = x− 1

and the following nine finite singularities:

M1(0, 0), M2(0,−1), M3(0,−1− 1/r), M4(1, 0), M5(1, 1),

M6(1,−1), M7(−1,−1), M8(1 + r, 0), M9(1/(1 + r),−1 + 1/(1 + r)).

It is easy to determine that 6 of these singularities are located at the intersections
of the above invariant lines, more precisely these are the singular points Mi for
i ∈ {1, 2, 4, 5, 6, 7}. The singular point M3 (respectively M8; M9) is located on the
invariant line L1 (respectively L3; L6 ). Moreover we have three singular points
located at the intersections of three invariant lines. More exactly L1, L3 and L5

intersect at the point M1; L1, L4 and L6 intersect at the point M2 and L2, L3 and
L6 intersect at the point M4.

To determine all the possible configurations for system (3.138) we have to ex-
amine the positions of the singularities M3 (located on the invariant line x = 0),
M8 (located on the invariant line y = 0) and M9 (located on the invariant line
x− y = 1) depending on the value of the parameter r.

Considering Notation 3.6 and the coordinates x8 = 1 + r, y3 = −(1 + r)/r and
x9 = 1/(1 + r) it is not too hard to detect the following implications:

(i) x8 < 0 ⇒ −1 < y3 < 0, x9 < 0 ⇒ M8 ≺ M1 ≺ M4, M2 ≺ M3 ≺ M1,
M9 ≺M2 ≺M4;

(ii) 0 < x8 < 1 ⇒ y3 > 0, x9 > 1 ⇒ M1 ≺ M8 ≺ M4, M2 ≺ M1 ≺ M3,
M2 ≺M2 ≺M9;

(iii) x8 > 1 ⇒ y3 < −1, 0 < x9 < 1 ⇒ M1 ≺ M4 ≺ M8, M3 ≺ M2 ≺ M1,
M2 ≺M9 ≺M4.

We observe that in the case (i) (respectively, case (ii); (iii)) we arrive at the config-
uration given by Figure 2(a) (respectively, Figure 2(b); Figure 2(c)). However it is
not too difficult to detect that each one of these three configurations is equivalent
(see Definition 1.3 to Config. 7.66).

Figure 2. Configurations of invariant lines of type (2,2,2)



76 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

3.5. Systems with configuration type T = (2, 2, 1, 1). In this subsection we
construct a cubic system with four real infinite singular points which has six invari-
ant affine straight lines with the configuration of the type T = (2, 2, 1, 1), having
total multiplicity seven, as always the invariant straight line at infinity is included.
According to Theorem 2.10 in this case necessarily the condition V5 = 0 holds.

3.5.1. Construction of the associated homogeneous cubic system. As a first step we
construct the associated to the system (3.1) homogeneous cubic system for which
the condition V5 = 0 is fulfilled. Since we have 4 real infinite distinct singularities,
according to Lemma 2.14 we consider the family of systems

ẋ =(p+ r)x3 + (s+ v)x2y + qxy2,

ẏ =px2y + (r + v)xy2 + (q + s)y3, rs(r + s) 6= 0,
(3.139)

and we shall force the condition V5 = 0 to be satisfied.
We observe that the invariant polynomial V5 is a homogeneous polynomial of

degree four in x and y. So we shall use the following notations:

V5 =

4∑
j=0

V5jx4−jyj .

Calculating the value of the polynomial V5 for system (3.38) we obtain

V51 = 128pq(pr+ 2r2− ps− rs+ rv)/9, V53 = −128pq(qr− qs+ rs− 2s2− sv)/9

and we consider two cases: pq 6= 0 and pq = 0. (1) Case pq 6= 0. Then V51 = V53 =

0 give us

p(r − s) + r(2r − s+ v) = 0 = q(r − s) + s(r − 2s− v).

(a) Subcase r − s 6= 0. In this case we obtain

p = (−2r2 + rs− rv)/(r − s), q = (−rs+ 2s2 + sv)/(r − s)
and this implies V5 = 0. Then after a time rescaling we obtain the family of systems

ẋ = r(r + v)x3 − (r − s)(s+ v)x2y + s(r − 2s− v)xy2,

ẏ = r(2r − s+ v)x2y − (r − s)(r + v)xy2 − s(s+ v)y3
(3.140)

with rs(r − s)(r + s)(2r − s+ v)(r − 2s− v) 6= 0 for which we calculate

H(X,Y, Z) = (r − s)X(X − Y )2Y (rX + sY )2.

So we observe that these system could have the two couple of parallel invariant
affine straight lines in the directions x− y = 0 and rx+ sy = 0. However it is more
convenient to have such lines in the directions x = 0 and y = 0. So applying the
change

(x, y, t) 7→
(rx+ sy)

r
, x− y, r

(r + s)(r − s)
t
)

systems (3.140) can be brought to the form

ẋ = x2
[
rx+

2rs+ rv + sv

r − s
y
]
, ẏ = y2

[ (r2 + s2 + rv + sv)

r − s
y + sy

]
.

Finally, since r−s 6= 0 setting a new parameter u = (r+s)(s+v)/(r−s) we obtain
the 3-parameter family of homogeneous systems:

ẋ = x2
[
rx+ (s+ u)y

]
, ẏ = y2

[
(r + u)x+ sy

]
(3.141)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 77

with rs(r + s)(r − s) 6= 0.

(b) Subcase r − s = 0. Setting s = r we calculate V51 = 128pqr(r + v)/9 and
since pqr 6= 0 we obtain v = −r. Then we obtain V52 = 64pqr(p + q + 3r)/3 = 0
which implies p = −(q + 3r). In this case we obtain V5 = 0 and applying the time
rescaling t→ −t we arrive at the following family of systems

ẋ = x
[
(q + 2r)x2 − qy2

]
, ẏ = y

[
(q + 3r)x2 − (q + r)y2

]
.

with qr(q + 3r) 6= 0. In this case we apply the transformation (x, y, t) 7→ (x + y,
y − x, t/(2r)) and we obtain the systems

ẋ = x2
[
x+ (2q + 3r)y/r

]
, ẏ = y2

[
(2q + 3r)x/r + y

]
.

Setting a new parameter u1 = (2q + 3r)/r we arrive at the systems

ẋ = x2(x+ u1y), ẏ = y2(u1x+ y), u1 6= 1.

We observe that these systems could be a subfamily of systems (3.141) if we allow
r = s. Indeed, setting s = r 6= 0 in systems (3.141) we may assume r = 1 (due to
the time rescaling t→ t/r) and then for u = u1 − 1 we obtain the above systems.

(2) Case pq = 0. Then without loss of generality we may assume p = 0 due
to the change (x, y, p, q, r, s, v) 7→ (y, x, q, p, s, r, v) which conserves systems (3.38).
For these systems for p = 0 we calculate

V50 = V51 = V52 = V53 = 0, V54 = −32q(q+ 2r+ 2s+v)(qr+ rs+ s2− sv)/9 = 0.

So we consider three subcases: (i) q = 0; (ii) q 6= 0 and v = −(q + 2r + 2s) and
(iii) q(q + 2r + 2s+ v) 6= 0 and v = (qr + rs+ s2)/s.

(a) Subcase q = 0. In this case we obtain the family of systems

ẋ = x2
[
rx+ (s+ v)y

]
, ẏ = y2

[
(r + v)x+ sy

]
with rs(r+s) 6= 0 which coincides with the family (3.141) (removing the restriction
r − s 6= 0).

(b) Subcase q 6= 0 and v = −(q + 2r + 2s). This leads to the systems

ẋ = x
[
rx2 − (q + 2r + s)xy + qy2

]
, ẏ = y2

[
− (q + r + 2s)x+ (q + s)y

]
with qrs(r + s) 6= 0. Applying the transformation (x, y, t) 7→ (x− y,−y,−t) and
setting three new parameters s1 = r + s, r1 = −r and u1 = −(q + 2s) (i.e. r =
−r1, s = r1 + s1, q = −2r1 − 2s1 − u1) we arrive at the family of systems

ẋ = x2
[
r1x+ (s1 + u1)y

]
, ẏ = y2

[
(r1 + u1)x+ s1y

]
which coincides with the family (3.141).

(c) Subcase q(q + 2r + 2s + v) 6= 0 and v = (qr + rs + s2)/s. Then we obtain
the systems

ẋ = x
[
rx2 + (qr/s+ r + 2s)xy + qy2

]
, ẏ = y2

[
(qr/s+ 2r + s)x+ (q + s)y

]
with qrs(r+s) 6= 0. Applying the transformation (x, y, t) 7→ ((x− y)/r, y/s, −rst)
we arrive at the family of systems

ẋ = x2
[
− sx+ (−qr − rs+ s2)y/s

]
, ẏ = y2

[
− (qr + 2rs+ s2)x/s+ (r + s)y

]
.

Then setting three new parameters

r2 = −s, s2 = r + s, u2 = −3(q + 2s)

s



78 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

⇒ r = r2 + s2, s = −r2, q =
r2(2r2 + 2s2 + u2)

r2 + s2

we obtain the family of systems

ẋ = x2
[
r2x+ (s2 + u2)y

]
, ẏ = y2

[
(r2 + u2)x+ s2y

]
which again coincides with the family (3.141).

Thus for further examination it remains only the family (3.141) with the con-
dition rs(r + s) 6= 0. Since rs 6= 0 in the generic systems with the associated
homogeneous cubic systems (3.141), we may consider g = n = 0 in system (2.4)
(due to a translation) and s = 1 (due to the time rescaling t→ t/s). So we obtain
the following family of cubic systems

ẋ = a+ cx+ dy + 2hxy + ky2 + rx3 + (1 + u)x2y,

ẏ = b+ ex+ fy + lx2 + 2mxy + (r + u)xy2 + y3, r(r + 1) 6= 0
(3.142)

which will be considered below.

3.5.2. Construction of a cubic system possessing invariant lines with configuration
type T = (2, 2, 1, 1). In what follows we shall determine necessary and sufficient
conditions for a system (3.142) to have invariant lines with the configuration of the
type T = (2, 2, 1, 1).

Considering Remark 2.13 for the homogeneous systems (3.141), associated to
system (3.142) we calculate

H(X,Y, Z) = gcd(G1,G2,G3) = X2Y 2(X − Y )(rX + Y ).

So each one of the invariant lines x = 0 and y = 0 (respectively, y = x and
y = −rx) of systems (3.141) is of multiplicity two (respectively, one). However for
some values of the parameters r and u the common divisor gcd(G1,G2,G3) could
contain additional factors (see Notation 2.4 and Lemma 2.6). We prove the next
lemma.

Lemma 3.17. For the existence of a couple of parallel invariant line in the direction
y = x (respectively, y = −rx) of a system (3.142) the condition 2(r + 1) + u = 0
(respectively r + 1− u = 0) is necessary.

Proof. We examine each one of the directions mentioned in the statement of the
lemma. (1) Direction y = x. Considering the equation (2.5) and Remark 2.12 for

system (3.142) we obtain

Eq6 = l − 2h− k + 2m− (2r + 2 + u)W,

Eq9 = e+ f − c− d+ (l + k)W + (1− r)W 2,

Eq10 = b− a+ (e− c)W + lW 2 − rW 3.

(3.143)

We observe that the equation Eq6 = 0 is of degree one with respect to W . So it is
clear that for the existence of two distinct solutions of the above equations (with
respect to W ), the condition 2(r+ 1) + u = 0 is necessary. This proves the validity
of the lemma concerning the direction y = x.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 79

(2) Direction y = −rx. In a similar way considering Remark 2.12 for system
(3.142) we calculate:

Eq6 = 2hr + 2m− (kr3 + l)/r + (r + 1− u)W,

Eq9 = dr − c+ f − e/r − (kr3 − l)
r2

W +
r − 1

r
W 2,

Eq10 = ar + b+
[
r(cr + e)W + lW 2 − rW 3

]
/r2.

(3.144)

We again observe that the equation Eq6 = 0 is of degree one with respect to W .
So for the existence of two distinct solutions of the above equations (with respect
to W ), the condition r + 1− u = 0 is necessary. This completes the proof. �

Lemma 3.18. A system (3.142) possesses one couple of parallel invariant lines in
the direction x = 0 if and only if the following conditions hold

1 + u 6= 0, k = 0,

d =
4h2r + c(1 + u)2

r(1 + u)
, a = −

2h
[
4h2r + c(1 + u)2

]
(1 + u)3

.
(3.145)

Systems (3.142) possesses one couple of parallel invariant lines in the direction
y = 0 if and only if

r+u 6= 0, l = 0, e =
4m2 + f(r + u)2

r + u
, b = −

2m
[
4m2 + f(r + u)2

]
(r + u)3

. (3.146)

Proof. We consider each one of the directions x = 0 and y = 0 and force the
existence of a couple of parallel invariant lines in each one of these directions.

(1) Direction x = 0. Considering the equations (2.5) and Remark 2.12 for system
(3.142) we obtain

Eq7 = k, Eq9 = d− 2hW + (1 + u)W 2, Eq10 = a− cW − rW 3.

Therefore the condition Eq7 = 0 gives us k = 0 and the equation Eq9 = 0 could
have two solutions only if 1 + u 6= 0. On the other hand since r(1 + u) 6= 0, by
Lemma 2.8 for the existence of two common solutions of the equations Eq9 = 0 and
Eq10 = 0 the following conditions are necessary and sufficient:

R
(0)
W (Eq9, Eq10) = R

(1)
W (Eq9, Eq10) = 0.

Calculations yield

R
(1)
W (Eq9, Eq10) = −c(1 + u)2 − r(4h2 − d− du) = 0

and this gives c = r(d+du−4h2)
(1+u)2 . Then we calculate

R
(0)
W (Eq9, Eq10) =

[
a(1 + u)2 + 2dhr]2/(1 + u) = 0

and we obtain a = −2dhr/(1 + u)2. For these values of the parameters c and a we
obtain

Eq9 = d−2hW + (1 +u)W 2, Eq10 = −r(2h+W + uW )

(1 + u)2
[
d−2hW + (1 +u)W 2

]
,

i.e. we have two common solutions, which could be real or complex, distinct or
coinciding.



80 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

(2) Direction y = 0. In this case considering Remark 2.12 for system (3.142)
calculations yield

Eq′5 = l, Eq′8 = e− 2mW + (r + u)W 2, Eq′10 = b− fW −W 3. (3.147)

Hence the condition Eq′5 = 0 gives us l = 0 and the equation Eq′8 = 0 could have
two solutions only if r + u 6= 0. Then by Lemma 2.8 for the existence of two
common solutions of the equations Eq′8 = 0 and Eq′10 = 0 the following conditions
are necessary and sufficient

R
(0)
W (Eq′8, Eq

′
10) = R

(1)
W (Eq′8, Eq

′
10) = 0.

We calculate

R
(1)
W (Eq′8, Eq

′
10) = −4m2 + e(r + u)− f(r + u)2 = 0

and this gives f = er+eu−4m2

(r+u)2 . Then calculations yield

R
(0)
W (Eq′8, Eq

′
10) =

[
b(r + u)2 + 2em]2/(r + u) = 0

and we obtain b = −2em/(r + u)2. For these values of the parameters f and b we
obtain

Eq′8 = e− 2mW + (r + u)W 2,

Eq′10 = −2m+ rW + uW

(r + u)2
[
e− 2mW + (r + u)W 2

]
,

i.e. we have two common solutions, which could be real or complex, distinct or
coinciding. This completes the proof of the lemma. �

Next we examine the remaining two directions. Taking into account Lemma 3.17
we consider two cases: (2r+ 2 + u)(r+ 1− u) 6= 0 and (2r+ 2 + u)(r+ 1− u) = 0.
Case (2r + 2 + u)(r + 1 − u) 6= 0. By Lemma 3.17 system (3.142) could possess a
couple of parallel invariant lines, neither in the direction y = x, nor in the direction
y = −rx. Therefore we could have two couples of such invariant lines: one in the
direction x = 0 and one in the direction y = 0. So assuming that the conditions
(3.145) and (3.146) are fulfilled (this guarantees the existence of the two couples of
parallel invariant lines), we have to force the existence of two invariant lines: one
in the direction y = x and the other in the direction y = −rx.

(a) Direction y = x. In this case we consider (3.143). Since (2 + 2r+u) 6= 0 the
equation Eq6 = 0 gives W = l−2h−k+2m

2+2r+u ≡W0 and we calculate

Eq9|{W=W0} =
H1(c, d, e, f, l, k,m, r, u)

(2 + 2r + u)2
,

Eq10|{W=W0} =
H2(c, d, e, f, l, k,m, r, u)

(2 + 2r + u)3
,

where H1 and H2 are the polynomials of degree 3 and 4 in the indicated parameters.
It is clear that we could have an invariant straight line in the direction y = x if and
only if H1 = H2 = 0.

(b) Direction y = −rx. We consider now the equations (3.144) and analogously
as above we detect that since (r + 1 − u) 6= 0 the equation Eq6 = 0 gives W =



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 81

− l−2mr−2hr
2+kr3

r(1+r−u) ≡W ′0. Herein considering (3.144) we calculate

Eq9|{W=W ′
0} =

H ′1(c, d, e, f, l, k,m, r, u)

r3(1 + r − u)2(1 + u)(r + u)
,

Eq10|{W=W ′
0} =

H ′2(c, d, e, f, l, k,m, r, u)

r(1 + r − u)3(1 + u)3(r + u)3
.

with some polynomials H ′1 and H ′2. So we deduce that the condition H ′1 = H ′2 = 0
is necessary and sufficient for the existence of exactly one invariant line in the
direction y = −rx.

Thus we conclude that to have invariant lines with the configuration of the type
T = (2, 2, 1, 1), we need to join the conditions (3.145), (3.146) and H1 = H2 = H ′1 =
H ′2 = 0. We stress that in the considered case for system (3.142) the additional
condition

r(r + 1)(u+ 1)(r + u)(2 + 2r + u)(r + 1− u) 6= 0 (3.148)

must hold.
Assume that the conditions (3.145) and (3.146) are satisfied. For these values

of the parameters a, b, d, e, k and l we evaluate the polynomials H1 and H ′1 which
turn out to be linear with respect to the parameters c and f . We obtain

H1 =
1

r
(1 + r + u)(2 + 2r + u)2(−c+ fr) +

4Φ1(h,m, r, u)

(1 + u)(r + u)
,

H ′1 = r2u(1 + r − u)2(cr − f)
4r2Φ′1(h,m, r, u)

(1 + u)(r + u)
,

where

Φ1 =2hm(1− r)(1 + u)(r + u)−m2(1 + u)(4 + 9r + 3r2 + 5u+ 3ru+ u2)

+ h2(r + u)(3 + 9r + 4r2 + 3u+ 5ru+ u2),

Φ′1 =2hm(−1 + r)r(1 + u)(r + u) +m2(1 + u)(−1− 3r + u+ 3ru− u2)

+ h2r2(r + u)(3r + r2 − 3u− ru+ u2).

We calculate

Coefficient[H1, c]× Coefficient[H ′1, f ]− Coefficient[H1, f ]× Coefficient[H ′1, c]

= ru(1− r)(1 + r + u)(1 + r)(1 + r − u)2(2 + 2r + u)2 ≡ ∆cf .

Clearly the equations H1 = 0 and H ′1 = 0 have a unique solution with respect to the
parameters c and f if and only if the condition ∆cf 6= 0. Considering the condition
(3.148) we deduce that this condition is equivalent to u(1 − r)(1 + r + u) 6= 0.
So in what follows we examine two possibilities: u(1 − r)(1 + r + u) 6= 0 and
u(1− r)(1 + r + u) = 0.

1. Possibility u(1 − r)(1 + r + u) 6= 0. Then solving the equations H1 = 0 and
H ′1 = 0 with respect to parameters c and f we obtain

c =
−4r

(1 + u)(r + u)∆cf

[
u(1 + r − u)2Φ1 − (1 + r + u)(2 + 2r + u)2Φ′1

]
,

f =
−4r2

(1 + u)(r + u)∆cf

[
r2(1 + r − u)2uΦ1 − (1 + r + u)(2 + 2r + u)2Φ′1

]
.

(3.149)



82 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

So considering these conditions as well as the conditions (3.145) and (3.146) we
calculate

H2 =
8r(1 + r)(2 + 2r + u)3

(1 + u)3(r + u)3∆cf
V1V2V3, H ′2 = −8r6(1 + r)(1 + r − u)3

(1 + u)3(r + u)3∆cf
V1V2V4,

where

V1 = h(2r − u)(r + u) +m(u− 2)(1 + u),

V2 = hr(r + u)(3 + r + u) +m(1 + u)(1 + 3r + u),

V3 = hr(r + u)(2r + 4r2 + 2r3 − u+ 2ru+ 3r2u− 8u2 + 2r2u2 − 4u3

+ 3ru3 + u4) +m(1 + u)(r3u− 2r − 4r2 − 2r3 − 3ru− 2r2u− 2u2

+ 8r2u2 − 3u3 + 4ru3 − u4),

V4 = h(r + u)(2r3 + 2r4 − 2r − 2r2 + u− 7ru− 9r2u− r3u+ u2

− 15ru2 − 10r2u2 − u3 − 8ru3 − u4) +m(1 + u)(2 + 2r − 2r2 − 2r3

− u− 9ru− 7r2u+ r3u− 10u2 − 15ru2 + r2u2 − 8u3 − ru3 − u4).

(3.150)

Therefore the equations H2 = 0 and H ′2 = 0 have a common solution if and only if
either V1 = 0 or V2 = 0 or V3 = V4 = 0. We examine each one of these cases.

1.1. Case V1 = 0. In this case we consider two subcases: 2r − u 6= 0 and
2r − u = 0.

1.1.1. Subcase 2r − u 6= 0. Since r + u 6= 0 (see the condition (3.148)) the

equation V1 = 0 implies h = m(2−u)(1+u)
(2r−u)(r+u) and we arrive at the family of systems:

ẋ =
[
x− 2m(2u+ ru− 1− 5r)

(1 + r − u)(2r − u)(r + u)

][
x+

2m(3r − 1 + u− u2)

(1 + r − u)(2r − u)(r + u)

]
×
[
rx+ (1 + u)y +

2mr(u− 2)

(2r − u)(r + u)

]
,

ẏ =
[
y +

2m(5r + r2 − u− 2ru)

(1 + r − u)(2r − u)(r + u)

][
y +

2m(r2 − 3r − ru+ u2)

(1 + r − u)(2r − u)(r + u)

]
×
[
(r + u)x+ y − 2m

r + u

]
.

(3.151)

To simplify these system we need to use a transformation which depends on two
possibilities: m(u2 + u+ ru− 8r) 6= 0 and m(u2 + u+ ru− 8r) = 0.

1.1.1.1. Possibility m(u2 + u + ru − 8r) 6= 0. In this case, because of the
condition (1 + r − u)(2r − u)(r + u) 6= 0 we may apply the transformation

x1 = αx− 1 + 5r − 2u− ru
u2 + u+ ru− 8r

, y1 = αy +
5r + r2 − u− 2ru

u2 + u+ ru− 8r
,

t1 =
4m2(u2 + u+ ru− 8r)2

(1 + r − u)2(2r − u)2(r + u)2
t, α = − (1 + r − u)(2r − u)(r + u)

2m(u2 + u+ ru− 8r)
.

This leads to the 2-parameter family of systems (we keep the old notations for the
variables)

ẋ = x(x− 1)
[
rx+ (1 + u)y − r − 1

]
,

ẏ = y(y − 1)
[
(r + u)x+ y − r − 1

]
,

(3.152)

for which the condition

ru(r2−1)(u+1)(r+u)(2+2r+u)(2r−u)(u2+u+ru−8r)
[
(r+1)2−u2

]
6= 0 (3.153)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 83

holds. These systems possess six distinct real invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : y = 0,

L4 : y = 1, L5 : y = x, L6 : rx+ y = r + 1

and the following nine finite singularities:

M1(0, 0), M2(0, 1), M3(0, 1 + r), M4(1, 0), M5(1, 1), M6(1, 1− u),

M7

(r + 1

r
, 0
)
, M8

(r − u
r

, 1
)
, M9

( r + 1

1 + r + u
,

r + 1

1 + r + u

)
.

(3.154)
We determine that due to the condition (3.153) all these singularities are distinct
if and only if (u− 1)(u− r) 6= 0. In the case u = 1 (respectively u = r) the singular
point M6 (respectively M8) coalesces with M4 (respectively M2).

It is easy to determine that six of these singularities are located at the intersec-
tions of the above invariant lines, more precisely, these are the singular points Mi

for i ∈ {1, 2, 3, 4, 5, 7}. The singular point M6 (respectively M8; M9) is located on
the invariant line L2 (respectively L4; L5). Moreover we have one singular point
located at the intersection of four invariant lines and one singularity is located at
the intersection of three invariant lines. More exactly L2, L4, L5 and L6 intersect
at the point M5 whereas L1, L3 and L5 intersect at the point M1.

To determine all the possible configurations for system (3.152) we have to ex-
amine the positions of the invariant lines as well as of the singularities M6, M8 and
M9 depending on the parameters r and u.

Let us first examine the position of the invariant lines. We observe that five
of the lines are fixed and only the position of the invariant line L6 depends on
the parameter r. Since this line is defined by y = −rx + r + 1, comparing with
other three invariant lines (L2, L4 and L5) which intersect L6 at the same point
M5(1, 1) we detect that we could have geometrically distinct situations in three
cases: (i) r < −1; (ii) − 1 < r < 0 and (iii) r > 0.

Next we consider the position of the finite singularity M6(1, y6) with y6 = 1− u
(respectively M8(x8, 1) with x8 = (r − u)/r; M9(x9, y9) with x9 = y9 = (r +
1)/(1 + r + u)) on the invariant line x = 1 (respectively y = 1; y = x) with respect
to the singular points M4(1, 0) and M5(1, 1) (respectively M2(0, 1) and M5(1, 1);
M1(0, 0) and M5(1, 1)). Considering Notation 3.6 and the coordinates y6 = 1− u,
x8 = (r − u)/r and x9 = (r + 1)/(1 + r + u) we have the next implications.

(I) For the singular point M6,

y6 ≤ 0 ⇒ M6 �M4 ≺M5; 0 < y6 < 1 ⇒ M4 ≺M6 ≺M5;

y6 > 1 ⇒ M4 ≺M5 ≺M6.

(II) For the singular point M8,

x8 ≤ 0 ⇒ M8 �M2 ≺M5; 0 < x8 < 1 ⇒ M2 ≺M8 ≺M5;

x8 > 1 ⇒ M2 ≺M5 ≺M8.

(III) For the singular point M9,

x9 < 0 ⇒ M9 ≺M1 ≺M5; 0 < x9 < 1 ⇒ M1 ≺M9 ≺M5;

x9 > 1 ⇒ M1 ≺M5 ≺M9.

So it is clear that the three possibilities mentioned before, for the position of
the invariant line L6, defined by the parameter r (i.e. r < −1, −1 < r < 0 and



84 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

r > 0), we have to confront with the possibilities for the three singularities M6 M8

and M9 mentioned above. Since we only have two parameters it is evident that
not all of the above possibilities are realizable. So examining the compatibilities
of the conditions it is not too hard to convince ourselves (using, for example, the
tools ”FindInstance” or ”Reduce” of computer algebra system Mathematica) that
the following lemma is valid.

Lemma 3.19. The family of systems (3.152) with the condition (3.153) possesses
the following configurations of invariant lines when the corresponding conditions
indicated below are satisfied:

Config. 7.67 ⇔


r < −1, u > 1, r > −1− u, (r = −2, u = 5/4);

or − 1 < r < 0, u < r < −1− u, (r = −1/8, u = −3/4);

Config. 7.68 ⇔ r < −1− u, u > 1, (r = −4, u = 2);

or u < r < 0, r > −1− u, (r = −7/8, u = 1/4);

Config. 7.69 ⇔

{
−2 < r < −1, u = 1, (r = −5/4, u = 1);

or − 1 < r < −1/2, u = r, (r = −5/6, u = −5/6);

Config. 7.70 ⇔

{
r < −2, u = 1, (r = −5/2, u = 1);

or − 1/2 < r < 0, u = r, (r = −1/4, u = −1/4);

Config. 7.71 ⇔

{
−1− u < r < −1, 0 < u < 1, (r = −11/8, u = 1/2);

or − 1 < r < u, u < −1− u, (r = −5/6, u = −7/12);

Config. 7.72 ⇔

{
r < −1− u, 0 < u < 1, (r = −2, u = 1/2);

or − 1− u < r < 0, u < 0, (r = −5/8, u = −1/8);

Config. 7.73 ⇔

{
u < r < −1, (r = −2, u = −3);

or − 1 < r < 0, u > 1, (r = −3/4, u = 2);

Config. 7.74 ⇔

{
r < −1, r < u < 0, (r = −4, u = −2);

or − 1 < r < 0, 0 < u < 1, (r = −7/8, u = 1/4);

Config. 7.75 ⇔

{
u = r < −1, (r = −2, u = −2);

or − 1 < r < 0, u = 1, (r = −1/2, u = 1);

Config. 7.7 ⇔ 0 < r < u, u > 1, (r = 1/8, u = 5/4);

Config. 7.77 ⇔

{
1 < u < r, (r = 2, u = 5/4);

or 0 < r < u, u < 1, (r = 1/16, u = 1/2);

Config. 7.78 ⇔

{
0 < r < 1, u = 1, (r = 1/7, u = 1);

or r = u > 1, (r = 2, u = 2);

Config. 7.79 ⇔

{
r > 1, u = 1, (r = 2, u = 1);

or 0 < r = u < 1, (r = 1/2, u = 1/2);

Config. 7.80 ⇔ r > u, 0 < u < 1, (r = 7/8, u = 1/2);

Config. 7.81 ⇔ 0 < r < −1− u, (r = 1/2, u = −5);

Config. 7.82 ⇔ r > 0, r > −1− u, u < 0, (r = 1/2, u = −5).



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 85

1.1.1.2. Possibility m(u2 + u+ ru− 8r) = 0.

1.1.1.2.1. Case m = 0. Then (3.151) become cubic homogeneous

ẋ = x2(rx+ y + uy), ẏ = y2(rx+ ux+ y), r(r + 1) 6= 0 (3.155)

possessing four distinct invariant affine lines: x = 0 (double), y = 0 (double), y = x
and y = −rx. We observe that the unique singularity (0, 0) of these systems is of
multiplicity 9. Since the invariant lines x = 0 and y = 0 are double and other two
are simple we deduce that we could have two geometrically distinct possibilities:
when the simple invariant lines are adjacent (if r < 0) and when they are not
adjacent (if r > 0). Therefore we obtain the configuration of invariant lines given
by Config. 7.83 if r < 0 and by Config. 7.84 if r > 0.

1.1.1.2.2. Case m 6= 0 and (u2+u+ru−8r) = 0. So we have r(u−8)+u(1+u) =
0 and clearly u 6= 8 (due to u(1+u) 6= 0). Therefore we obtain r = u(1+u)/(8−u)
and system (3.151) becomes

ẋ =
(1 + u)

[
m(u− 8)2 − 27u2x

]2
19683(8− u)u5

[
2m(u− 8)2 + 27u2x− 27(u− 8)uy

]
,

ẏ =
1

729(u− 8)u3
(mu− 8m− 9uy)2

[
2m(u− 8)2 − 81u2x+ 9(u− 8)uy

]
.

(3.156)

Since u 6= 0 applying the transformation

x1 = x− m(u− 8)2

27u2
, y1 = y − m(u− 8)

9u
, t1 =

t

8− u
we arrive at the 1-parameter family of systems (we keep the old notations for
variables)

ẋ = (1 + u)x2
[
ux+ (8− u)y

]
, ẏ = y2

[
9ux+ (8− u)y

]
. (3.157)

On the other hand, setting in systems (3.155) the condition r = u(1 + u)/(8 − u)
we obtain the 1-parameter family of systems

ẋ = (1 + u)x2
[
ux+ (8− u)y

]
/(8− u), ẏ = y2

[
9ux+ (8− u)y

]
/(8− u).

Evidently applying the time rescaling t → (8 − u)t we obtain systems (3.157) and
hence we could not have new configurations.

1.1.2. Subcase 2r − u = 0. Then u = 2r and considering (3.150) the condition
V1 = 0 becomes V1 = 2m(r − 1)(1 + 2r) = 0. Since by the condition (3.148) we
have (u+ 1)(r + 1− u) = (2r + 1)(1− r) 6= 0 the condition V1 = 0 implies m = 0.
This leads to the family of systems

ẋ =
(
x− h

r − 1

)[
x+

h(−1 + 4r)

(r − 1)(1 + 2r)

][
rx+ (1 + 2r)y − 2hr

1 + 2r

]
,

ẏ =
[
y − 3hr

(r − 1)(1 + 2r)

][
y +

3hr

(r − 1)(1 + 2r)

](
3rx+ y

)
.

(3.158)

1.1.2.1. Possibility h 6= 0. Then we can apply the transformation

x1 =
(1− r)(1 + 2r)

6hr
x+

1 + 2r

6r
, y1 =

(1− r)(1 + 2r)

6hr
y + 1/2,

t1 =
36h2r2t

(r − 1)2(1 + 2r)2
.



86 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

and this leads to the systems (we keep the old notations for variables)

ẋ = x(x− 1)
[
rx+ (1 + 2r)y − r − 1

]
, ẏ = y(y − 1)

[
3rx+ y − r − 1

]
.

We observe that after the translation y → y + r these systems become a subfamily
of the family (3.152) defined by the condition u = 2r and hence the corresponding
configurations are already determined.

1.1.2.2. Possibility h = 0. In this case system (3.158) becomes homogeneous
system (3.155) with u = 2r, i.e. we obtain Config. 7.83 if r < 0 and by Config. 7.84
if r > 0.

1.2. Case V1 6= 0 and V2 = 0. Considering (3.150) we obtain the condition

V2 = hr(r + u)(3 + r + u) +m(1 + u)(1 + 3r + u) = 0

and as r(r+u) 6= 0 (see the condition (3.148)) we examine two subcases: 3+r+u 6= 0
and 3 + r + u = 0.

1.2.1. Subcase 3 + r + u 6= 0. In this case the condition V2 = 0 gives

h = −m(1 + u)(1 + 3r + u)

r(r + u)(3 + r + u)

and considering also the conditions (3.149), (3.145) and (3.146) we arrive at the
system

ẋ =
[
x− 2m(1 + 8r + 3r2 + u+ 2ru)

r(r + u)(3 + r + u)(2 + 2r + u)

]
×
[
x+

2m(1 + 3r2 + 2u+ 3ru+ u2)

r(r + u)(3 + r + u)(2 + 2r + u)

]
×
[
rx+ (1 + u)y +

2m(1 + 3r + u)

(r + u)(3 + r + u)

]
,

ẏ =
[
y +

2m(3 + 8r + r2 + 2u+ ru)

(r + u)(3 + r + u)(2 + 2r + u)

][
(r + u)x+ y − 2m

r + u

]
×
[
y +

2m(3 + r2 + 3u+ 2ru+ u2)

(r + u)(3 + r + u)(2 + 2r + u)

]
.

(3.159)

We again have to examine two possibilities: m(u2 + u + ru − 8r) 6= 0 and
m(u2 + u+ ru− 8r) = 0.

1.2.1.1. Possibility m(u2 + u+ ru− 8r) 6= 0. In this case due to the condition
r(r + u)(3 + r + u)(2 + 2r + u) 6= 0 we apply the transformation

x1 = αx− 1 + 8r + 3r2 + u+ 2ru

u2 + u+ ru− 8r
, y1 = αy +

r(3 + 8r + r2 + 2u+ ru)

u2 + u+ ru− 8r
,

t1 =
4m2(u2 + u+ ru− 8r)2t

r2(r + u)2(3 + r + u)2(2 + 2r + u)2
, α =

r(r + u)(3 + r + u)(2 + 2r + u)

2m(u2 + u+ ru− 8r)
.

This leads to the 2-parameter family of systems

ẋ = x(x− 1)
[
r1x+ (1 + u1)y − r1(1 + r1)

]
,

ẏ = y(y + r1)
[
(r1 + u1)x+ y + 1 + r1

]
.

However we detect, that setting r1 = 1/r, u1 = −(1 + r + u)/r and applying the
rescaling (x, y, t) 7→ (x,−y/r, r2t) we arrive at systems (3.152) for which all possible
configurations of invariant lines are provided by Lemma 3.19.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 87

1.2.1.2. Possibility m(u2 + u+ ru− 8r) = 0. The straightforward calculations
give us that for m = 0 system (3.159) becomes the cubic homogeneous system
(3.155), whereas for r = u(1 + u)/(8 − u) system (3.159) becomes exactly the
system (3.156), i.e. we do not have new configurations.

1.2.2. Subcase 3 + r + u = 0. Then u = −(3 + r) and considering (3.150) the
condition V2 = 0 becomes V2 = 2m(1−r)(2+r) = 0. Since by the condition (3.148)
we have (u+1)(2+2r+u) = (1−r)(2+r) 6= 0 we deduce that the condition V2 = 0
implies m = 0. This leads to the family of systems

ẋ =
(
x− h

r − 1

)[
x− h(r − 4)

(r − 1)(2 + r)

][
rx− (2 + r)y +

2hr

2 + r

]
,

ẏ =
[
y − 3hr

(r − 1)(2 + r)

][
y +

3hr

(r − 1)(2 + r)

](
y − 3x

)
.

(3.160)

1.2.2.1. Possibility h 6= 0. Then we can apply the transformation

x1 =
(1− r)(2 + r)

6h
x+

2 + r

6
, y1 =

(1− r)(2 + r)

6h
y+

r

2
, t1 =

36h2t

(r − 1)2(2 + r)2
.

and this leads to the systems (we keep the old notations for the variables)

ẋ = x(x− 1)
[
rx− (2 + r)y + r

]
, ẏ = y(y − r)

[
y − 3x+ 1

]
We observe that applying the translation y → y + r this system becomes a sub-
family of the family (3.152) defined by the condition u = −(3 + r) and hence their
corresponding configurations are already determined.

1.2.2.2. Possibility h = 0. Setting this condition in (3.160) we arrive at the
homogeneous system (3.155) with u = −(3 + r), i.e. we could not have new config-
urations.

1.3. Case V3 = V4 = 0. Considering (3.150) we observe that both polynomials
V3 and V4 are linear and homogeneous with respect to the parameters h and m.
Moreover calculations yield

Coefficient[V3, h]× Coefficient[V4,m]− Coefficient[V3,m]× Coefficient[V4, h]

= (1 + r)(1 + r − u)u(1 + u)2(r + u)2(1 + r + u)(2 + 2r + u)(u2 + u+ ru− 8r)

≡ ∆hm.

We observe that ∆hm 6= 0 due to the condition (3.148) and ∆cf 6= 0. Hence we
detect that the unique solution of the equations V3 = 0 and V4 = 0 is h = m = 0.

Therefore considering the conditions (3.149), (3.145) and (3.146), systems (3.142)
become homogeneous systems (3.155), possessing two configurations given by Con-
fig. 7.83 or Config. 7.84.

2. Possibility u(1−r)(1+r+u) = 0. In this case ∆cf = 0 and we have to return
and impose the conditions H1 = H2 = H ′1 = H ′2 = 0 to be satisfied. We examine
each one of three conditions: (i) u = 0; (ii) u 6= 0 and r = 1 and (iii) u(1− r) 6= 0
and (1 + r + u) = 0.

2.1. Case u = 0. Then we obtain

H ′1 = −4r(m− hr2)
[
hr2(3 + r) +m(1 + 3r)

]
= 0 (3.161)

and since r 6= 0 we have to examine two subcases: m − hr2 = 0 and hr2(3 + r) +
m(1 + 3r) = 0. We claim that the first condition leads to degenerate systems.



88 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Indeed, assume m = hr2. Then calculations yield

H ′1 = H ′2 = 0, H1 = 4(1 + r)3(−c+ fr − 3h2r + 3h2r3)/r,

H2 = 8h(r − 3)(1 + r)3(−c+ fr − 3h2r + 3h2r3).

Since by condition (3.148) we have r(r + 1) 6= 0, the equation H1 = 0 gives c =
r(f − 3h2 + 3h2r2). Considering the conditions u = 0 and m = hr2 and this value
of the parameter c we arrive at the systems

ẋ = −(2hr − rx− y)(f + h2 + 3h2r2 + 2hx+ x2),

ẏ = −(2hr − rx− y)(f + 4h2r2 + 2hry + y2)

which evidently are degenerate. So our claim is proved and we need to examine the
condition hr2(3 + r) + m(1 + 3r) = 0, considering two subcases: 1 + 3r 6= 0 and
1 + 3r = 0.

2.1.1. Subcase 1 + 3r 6= 0. Then m = −hr2(3 + r)/(1 + 3r) and we calculate

H1 =
4(1 + r)2

r

[
(1 + r)(fr − c) +

h2r(r − 1)

(1 + 3r)2
(3 + 24r + 58r2 + 24r3 + 3r4)

]
.

So the condition H1 = 0 gives

c = fr +
h2r(r − 1)

(1 + r)(1 + 3r)2
(
3 + 24r + 58r2 + 24r3 + 3r4

)
and therefore,

H2 =
64hr(1 + r)2

(1 + 3r)3
[
f(1 + r)2(1 + 3r)2 + h2r2(27 + 72r + 82r2 + 24r3 + 3r4)

]
= 0.

Since r(r + 1) 6= 0 and h 6= 0 (otherwise we obtain m = 0 and this leads to
degenerate systems) the condition H2 = 0 yields

f = −h
2r2(27 + 72r + 82r2 + 24r3 + 3r4)

(1 + r)2(1 + 3r)2
.

This implies H ′2 = 0 and we obtain the following family of systems

ẋ =
[
x+

h(1 + 3r2)

(1 + r)(1 + 3r)

][
x+

h(1 + 8r + 3r2)

(1 + r)(1 + 3r)

](
rx+ y − 2hr

)
,

ẏ =
[
y − hr(3 + r2)

(1 + r)(1 + 3r)

][
y − hr(3 + 8r + r2)

(1 + r)(1 + 3r)

][
rx+ y +

2hr(3 + r)

1 + 3r

]
.

Then setting r = 1/r1 and applying the transformation

x1 =
(1 + r1)(3 + r1)

8hr1
x− 3 + 8r1 + r21

8r1
,

y1 = − (1 + r1)(3 + r1)

8h
y +

1 + 8r1 + 3r21
8r1

, t1 =
64h2 t

(3 + 4r1 + r21)2

we arrive at the 1-parameter family of systems (we keep the old notations)

ẋ = x(x− 1)(rx− ry − 1− r),
ẏ = y(y − 1)(y − x− 1− r), r(r + 1)(3r + 1) 6= 0.

(3.162)

We observe that this family is a subfamily of systems (3.152) defined by the
condition u = −(1 + r), i.e. it corresponds to the case r + u + 1 = 0. On the
other hand for system (3.152) all possible configurations of invariant lines were



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 89

constructed provided that the condition (3.153) is satisfied. This condition includes
in particular the condition r + u + 1 6= 0. So considering the singularities (3.154)
of system (3.152) we deduce that the condition u = −(1 + r) forces the finite
singularity M9 to coalesce with the infinite singularity N [1 : 1 : 0].

We observe that the invariant lines of systems (3.162) coincide with those of
systems (3.152) (since they do not depend on parameter u):

L1 : x = 0, L2 : x = 1, L3 : y = 0,

L4 : y = 1, L5 : y = x, L6 : rx+ y = r + 1,

but system (3.162) has only eight finite singularities:

M1(0, 0), M2(0, 1), M3(0, 1 + r), M4(1, 0), M5(1, 1),

M6(1, 2 + r), M7

(r + 1

r
, 0
)
, M8

(2r + 1

r
, 1
)
.

We observe that because of the condition r(r + 1) 6= 0 all these singularities are
distinct if and only if (r+2)(2r+1) 6= 0. In the case r = −2 (respectively r = −1/2)
the singular point M6 (respectively M8) coalesces with M4 (respectively M2).

We determine again that 6 of these singularities are located at the intersections
of the above invariant lines, more precisely, these are the singular points Mi for
i ∈ {1, 2, 3, 4, 5, 7}. The singular point M6 (respectively M8) is located on the
invariant line L2 (respectively L4). Moreover we have one singular point located
at the intersection of four invariant lines and one singularity is located at the
intersection of three invariant lines. More exactly L2, L4, L5 and L6 intersect at
the point M5 whereas L1, L3 and L5 intersect at the point M1.

To determine all possible configurations of invariant lines for system (3.162) we
examine the positions of the invariant lines as well as of the singularities M6 and
M8 depending on the parameter r.

Since the invariant lines of systems (3.162) coincide exactly with those of system
(3.152), as it was mentioned, for the invariant lines of system (3.152) (see page 83),
in order to detect the position of the invariant line L6 (which is the only line which
depends on the parameter r) it is necessary to examine three cases: (i) r < −1;
(ii) − 1 < r < 0 and (iii) r > 0.

In addition we need to examine the position of the finite singularity M6(1, y6)
with y6 = 2 + r (respectively M8(x8, 1) with x8 = (2r + 1)/r) on the invariant
line x = 1 (respectively y = 1) with respect to the singular points M4(1, 0) and
M5(1, 1) (respectively M2(0, 1) and M5(1, 1)). Considering Notation 3.6 a we have
the following implications.

(I) For the singular point M6:

y6 ≤ 0 ⇒ M6 �M4 ≺M5; 0 < y6 < 1 ⇒ M4 ≺M6 ≺M5;

y6 > 1 ⇒ M4 ≺M5 ≺M6.

(II) For the singular point M8:

x8 ≤ 0 ⇒ M8 �M2 ≺M5; 0 < x8 < 1 ⇒ M2 ≺M8 ≺M5;

x8 > 1 ⇒ M2 ≺M5 ≺M8.

Considering the values of the coordinates y6 and x8 we obtain

y6 ≤ 0 ⇔ r ≤ −2; 0 < y6 < 1 ⇔ −2 < r < −1; y6 > 1 ⇔ r > −1;

x8 ≤ 0 ⇔ −1/2 ≤ r < 0; 0 < x8 < 1 ⇔ −1 < r < −1/2;



90 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

x8 > 1 ⇔ r < −1 or r > 0.

Taking into account the three possibilities mentioned before for the position of
the invariant line L6 defined by the parameter r (i.e. r < −1, −1 < r < 0 and
r > 0) it is not too difficult to convince ourselves that the following lemma is valid.

Lemma 3.20. The family of systems (3.162) with the condition r(r + 1)(3r +
1) 6= 0 possesses the following configurations of invariant lines if and only if the
corresponding conditions indicated below are satisfied:

Config. 7.85 ⇔ r < −2 or − 1/2 < r < 0, r 6= −1/3;

Config. 7.86 ⇔ r ∈ {−2,−1/2};
Config. 7.87 ⇔ −2 < r < −1/2, r 6= −1;

Config. 7.88 ⇔ r > 0.

2.1.2. Subcase 1 + 3r = 0. Then r = −1/3 and the condition hr2(3 + r) +m(1 +
3r) = 0 (see (3.161)) gives h = 0. Then we calculate

H ′1 = 0, H1 = 16(6c+ 2f − 27m2) = 0 ⇒ f = −3(2c− 9m2)/2

and this implies

H2 = −8m(16c− 243m2)/3 = 0, H ′2 = −4(16c− 243m2)/729 = 0.

Since m 6= 0 (otherwise we obtain m = h = 0 and this leads to degenerate system),
the condition c = 243m2/16 must hold. In this case we arrive at the family of
systems

ẋ = (27m− 4x)(27m+ 4x)(x− 3y)/48,

ẏ = (3m− 4y)(21m− 4y)(18m− x+ 3y)/48

with m 6= 0. Then applying the transformation

x1 =
2

27m
x+ 1/2, y1 =

2

9m
y − 1/6, t1 = 81m2t/4

we obtain the system (we keep the old notations)

ẋ = x(x− 1)(3y − 3x+ 2), ẏ = y(y − 1)(y − x+ 2).

We observe that this system belongs to the family (3.162) for r = −1/3 and by
Lemma 3.20 its configuration corresponds to Config. 7.85. So we can remove from
the lemma the condition r 6= −1/3, which corresponds to this configuration.

2.2. Case u 6= 0, r = 1. We detect that in this case for the system (3.142) the
following condition must hold:

r(r+1)(u+1)(r+u)(2r+2+u)(r+1−u) 6= 0 ⇒ (u+1)(u+4)(u−2) 6= 0 (3.163)

For r = 1 we calculate

H ′1 = (u− 2)2
[
u(c− f) + 4(h2 −m2)

]
/(1 + u)

and as u(u − 2) 6= 0 the condition H ′1 = 0 gives f = 4h2−4m2+cu+cu2

u(1+u) . In this case

calculations yield

H1 = 8(h−m)(h+m)(4 + u)2/(u(1 + u)) = 0.

Considering condition (3.163) we obtain (h−m)(h+m) = 0.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 91

2.2.1. Subcase h = m. In this case we have H1 = H ′1 = H2 = 0 and

H ′2 =
4mu(4 + u)

(1 + u)3
[
c(u− 2)2(1 + u)2 + 4m2(7− u+ u2)

]
= 0 (3.164)

and we consider two possibilities: m = 0 and m 6= 0.

2.2.1.1. Possibility m = 0. Then H ′2 = 0 and we arrive at the following family
of systems:

ẋ = (c+ x2)(x+ y + uy), ẏ = (c+ y2)(x+ ux+ y). (3.165)

We observe that for c = 0 these systems become homogeneous and belong to the
family (3.155) for r = 1.

Assume that condition c 6= 0 holds. Since the invariant lines c + x2 = 0 and
c + y2 = 0 could be real or complex depending on the sign of the parameter c we
examine both cases.

2.2.1.1.1. Case c < 0. Then setting c = −v2 and applying the transformation
(x, y, t) 7→

(
vx, vy, t/v2

)
we arrive at the 1-parameter family of systems

ẋ = (x2 − 1)
[
x+ (1 + u)y

]
, ẏ = (y2 − 1)

[
(1 + u)x+ y

]
(3.166)

with condition (3.163). These systems possess six distinct fixed invariant affine
straight lines

L1 : x = 1, L2 : x = −1, L3 : y = 1, L4 : y = −1, L5 : y = x, L6 : y = −x
and the following nine finite singularities:

M1(0, 0), M2(−1,−1), M3(−1, 1), M4(1,−1), M5(1, 1),

M6(−1, 1 + u), M7(1 + u,−1), M8(1,−1− u), M9(−1− u, 1).
(3.167)

We observe that the above systems are symmetric with respect to the origin of coor-
dinates, because of the change (x, y)→ (−x,−y). The systems are also symmetric
with respect to the invariant line y = x, because the change (x, y) → (y, x) also
conserves these system. As a consequence we have two pairs of finite singularities
depending on the parameter u which are symmetric with respect to (0, 0) (M7 with
M9 and M6 with M8), and two pairs of singularities symmetric with respect to the
line y = x (M6 with M7 and M8 with M9).

Thus, because of the symmetry, it is sufficient to determine the position of the
singularity M6 with respect to the singular points M2 and M3 located on the same
invariant line x = −1. Moreover we observe that M6 could coalesce with M2

for u = −2 and due to the symmetry, M7 could also coalesce with M2 getting a
triple singular point M2 ≡ M6 ≡ M7. Clearly due to the symmetry we obtain
simultaneously a triple singular point M5 ≡M8 ≡M9.

Thus for the singular point M6(−1, 1 + u) we have (see Notation 3.6):

u ≤ −2 ⇒ M6 �M2 ≺M3; −2 < u < 0 ⇒ M2 ≺M6 ≺M3;

u > 0 ⇒ M2 ≺M3 ≺M6.

Examining the positions of all the singularities (3.167) corresponding to these con-
ditions we conclude that the following lemma is valid.

Lemma 3.21. The family of systems (3.166) with the condition u(u + 1) 6= 0
possesses the following configurations of invariant lines when the corresponding
conditions indicated below are satisfied:

Config. 7.89 ⇔ u < −2 or u > 0;



92 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Config. 7.90 ⇔ u = −2;

Config. 7.91 ⇔ −2 < u < 0;

2.2.1.1.2. Case c > 0. Then setting c = v2 and applying the same transforma-
tion (x, y, t) 7→

(
vx, vy, t/v2

)
we obtain the 1-parameter family of systems

ẋ =(x2 + 1)
[
x+ (1 + u)y

]
,

ẏ =(y2 + 1)
[
(1 + u)x+ y

]
, u(u+ 1)(u+ 4)(u− 2) 6= 0.

These systems possess two real and four complex invariant lines

y = ±x, x = ±i, y = ±i
as well as nine distinct finite singularities among which only one is real, namely
(0, 0). As a result we obtain the configuration Config. 7.92.

2.2.1.2. Possibility m 6= 0. In this case considering (3.164) the condition H ′2 = 0
gives

c = −4m2(7− u+ u2)

(u− 2)2(1 + u)2

and therefore we arrive at the family of systems

ẋ =
[
x− 6m

(u− 2)(1 + u)

][
x+

2m

u− 2

][
x+ (1 + u)y − 2m

u+ 1

]
,

ẏ =
[
y − 6m

(u− 2)(1 + u)

][
y +

2m

u− 2

][
(1 + u)x+ y − 2m

u+ 1

]
.

As m 6= 0 we can apply the transformation

x1 =
(u− 2)(1 + u)

2m(4 + u)
x+

u+ 1

u+ 4
,

y1 =
(u− 2)(1 + u)

2m(4 + u)
y +

u+ 1)

u+ 4
, t1 =

4m2(4 + u)2 t

(u− 2)2(1 + u)2

and we arrive at the 1-parameter family of systems (we keep the old notation)

ẋ =x(x− 1)
[
x+ (1 + u)y − u

]
,

ẏ =y(y − 1)
[
(1 + u)x+ y − u

]
, u(u+ 1)(u+ 4)(u− 2) 6= 0.

(3.168)

These systems possess six distinct fixed invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : y = 0, L4 : y = 1, L5 : y = x, L6 : y = −x
and the following nine finite singularities:

M1(0, 0), M2(0, 1), M3(0, u), M4(1, 0), M5(1, 1),

M6(1,−1), M7(−1, 1), M8(u, 0), M9

(
u/(2 + u), u/(2 + u)

)
.

(3.169)

It is clear that for u = 1 the singularity M3 (respectively M8) coalesces with M2

(respectively M4) producing two double finite singularities. Moreover if u = −2 the
singularity M9 coalesces with the singularity at infinity N [1 : 1 : 0].

We observe that the above systems are symmetric with respect to the invariant
line y = x, because the change (x, y)→ (y, x) conserves these system. As a conse-
quence we have one pair of finite singularities depending on the parameter u which
are symmetric with respect the line y = x: M3 and M8.

On the other hand due to the symmetry it is sufficient to determine the position
of the singularity M3 with respect to the singular points M1 and M2 located on the



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 93

same invariant line x = 0 as well as the position of the singularity M9 with respect
to the singular points M1 and M5 located on the same invariant line y = x.

So examining the positions of the singular points M3(0, u) and M9

(
u/(2 +

u), u/(2 + u)
)

we obtain, respectively

u < 0⇒M3 ≺M1 ≺M2; 0 < u < 1⇒M1 ≺M3 ≺M2;

u ≥ 1⇒M1 ≺M2 �M3,

and

u = −2⇒M9 ≡ N [1 : 1 : 0]; −2 < u < 0⇒M9 ≺M1 ≺M5;

u > 0⇒M1 ≺M9 ≺M5; u < −2⇒M1 ≺M5 ≺M9.

So considering the positions of all the singularities (3.169) corresponding to these
conditions we conclude that the following lemma is valid.

Lemma 3.22. The family of systems (3.168) with the condition u(u + 1)(u +
4)(u − 2) 6= 0 possesses the following configurations of invariant lines when the
corresponding conditions indicated below are satisfied:

Config. 7.76 ⇔ u > 1;

Config. 7.80 ⇔ 0 < u < 1;

Config. 7.81 ⇔ u < −2;

Config. 7.82 ⇔ −2 < u < 0;

Config. 7.88 ⇔ u = −2;

Config. 7.93 ⇔ u = 1.

2.2.2. Subcase h = −m. In this case we have

H1 = H ′1 = H ′2 = 0, H2 =
4m(u2 − 4)

(1 + u)3
[
c(1+u)2(4+u)2+4m2(13+5u+u2)

]
= 0.

If m = 0 we obtain h = 0 and this case was considered above and leads to systems
(3.165) and therefore we assume m 6= 0. On the other hand by the condition (3.163)
we have (1 + u)(4 + u)(u− 2) 6= 0. So the condition H2 = 0 gives

c = −4m2(13 + 5u+ u2)

(1 + u)2(4 + u)2

and we arrive at the family of systems

ẋ =
[
x− 2m

4 + u

][
x− 6m

(1 + u)(4 + u)

][
x+ (1 + u)y +

2m

u+ 1

]
,

ẏ =
[
y +

2m

4 + u

][
y +

6m

(1 + u)(4 + u)

][
(1 + u)x+ y − 2m

u+ 1

]
.

(3.170)

As m 6= 0 we can apply the transformation

x1 = − (1 + u)(4 + u)

2m(u− 2)
x+

u+ 1

u− 2
,

y1 = − (1 + u)(4 + u)

2m(u− 2)
y − u+ 1

u− 2
, t1 =

4m2(u− 2)2 t

(1 + u)2(4 + u)2



94 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

and this leads to the 1-parameter family of systems (we keep the old notation)

ẋ =x(x− 1)
[
x+ (1 + u)y + u+ 2

]
,

ẏ =y(y + 1)
[
(1 + u)x+ y − u− 2

]
, u(u+ 1)(u+ 4)(u− 2) 6= 0.

(3.171)

We observe that via the change (x, y, t, u) 7→ (x,−y, t,−u − 2) the above systems
can be brought to the systems (3.168) and hence no new configurations could be
obtained.

2.3. Case u(r − 1) 6= 0 and 1 + r + u = 0. In this case the following condition
must hold for system (3.142):

r(r + 1)(u+ 1)(r + u)(2r + 2 + u)(r + 1− u)u(r − 1) 6= 0 ⇒ u(u+ 1)(u+ 2) 6= 0.

For r = −u− 1 calculations yield

H1 = 4(h−m−mu)(2h− 2m+ 3hu−mu+mu2)/(1 + u) = 0

and we consider two subcases: h−m−mu = 0 and h−m−mu 6= 0.

2.3.1. Subcase h−m−mu = 0. Then h = m(1 + u) and we calculate

H1 = H2 = 0, H ′1 = 4u3(1 + u)2
[
− f − c(1 + u) + 6m2u+ 3m2u2

]
= 0.

Therefore from u(u+1) 6= 0 we obtain f = −c(1+u)+6m2u+3m2u2. This implies
H ′2 = 0 and we arrive at the family of degenerate system,

ẋ =(2m− x+ y)
[
− c+ 4m2 + 4m2u+ 2m(1 + u)x+ (1 + u)x2

]
,

ẏ =(2m− x+ y)
[
− c+ 4m2 − cu+ 6m2u+ 3m2u2 − 2my + y2

]
.

2.3.2. Subcase h−m−mu 6= 0. Then the condition H1 = 0 yields

h(2 + 3u) +m(u− 2)(1 + u) = 0. (3.172)

2.3.2.1 Possibility 2 + 3u = 0. Then u = −2/3 and this implies m = 0. In this
case we have

H1 = 0, H ′1 = 16(2c+ 6f + 9h2)/729 = 0

and this gives f = −(2c+ 9h2)/6. Then calculation yields

H1 = H ′1 = 0, H2 = 4h(16c− 171h2)/27 = 0, H ′2 = 8h(16c− 171h2))/2187 = 0.

If h = 0 we obtain the degenerate systems

ẋ = (3c− x2)(x− y)/3, ẏ = (x− y)(c− 3y2)/3.

So h 6= 0 and we must have c = 171h2/16. In this case we arrive at the family of
systems

ẋ = (3h+ 4x)(21h+ 4x)(6h− x+ y)/48, ẏ = (9h− 4y)(x− y)(9h+ 4y)/16

which via the transformation

x1 = −2x/(9h)− 1/6, y1 = −2y/(9h)− 1/2, t1 = −27h2t/4

can be brought to the system

ẋ = x(x− 1)(1 + x− y), ẏ = y(y + 1)(3x− 3y − 1).

This system possesses six distinct fixed invariant affine straight lines

L1 : x = 0, L2 : x = 1, L3 : y = 0,

L4 : y = −1, L5 : y = x− 1, L6 : x− 3y = 1



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 95

and eight finite singularities:

M1(0, 0), M2(0,−1), M3(0,−1/3), M4(1, 0),

M5(1,−1), M6(1, 2/3), M7(−1, 0), M8(−2,−1).

Considering Lemma 2.7 we calculate µ0 = 0 and µ1 = −36(x − y) 6= 0 and we
deduce that one finite singular point has coalesced with N [1 : 1 : 0]. As a result we
arrive at the configuration of invariant lines equivalent to Config. 7.85.

2.3.2.2. Possibility 2 + 3u 6= 0. Then from the condition (3.172) we obtain
h = −m(−2 + u)(1 + u)/(2 + 3u) and calculations yield

H1 = 0, H2 = − 8mu3

(2 + 3u)3

[
(2 + 3u)2(c+ f + fu) + 24m2u(1 + u)(2 + u)

]
= 0.

Since u 6= 0 we have to consider two cases: m = 0 and m 6= 0.

2.3.2.2.1. Case m = 0. This implies h = 0 and we obtain

H1 = H2 = H ′2 = 0, H ′1 = −4u3(1 + u)2(c+ f + cu) = 0.

Therefore from u(u+1) 6= 0 we obtain f = −c(1+u) and this leads to the degenerate
system

ẋ = (c− x2 − ux2)(x− y), ẏ = (x− y)(c+ cu− y2).

2.3.2.2.2. Case m 6= 0. Since u 6= 0 the condition H2 = 0 implies

f = −c(2 + 3u)2 + 24m2u(1 + u)(2 + u)

(1 + u)(2 + 3u)2

and then we have H1 = H2 = 0 and

H ′1 =
4u2(1 + u)(2 + u)

(2 + 3u)2

[
−cu2(2+3u)2+m2(1+u)(16+32u+28u2−12u3+3u4)

]
= 0.

Because u(1 + u)(2 + u) 6= 0 and 2 + 3u 6= 0, the condition H ′1 = 0 gives

c =
m2(1 + u)(16 + 32u+ 28u2 − 12u3 + 3u4)

u2(2 + 3u)2
.

This implies H ′2 = 0 and we arrive at the family of systems

ẋ =− (1 + u)
[
x− m(u2 − 6u− 4)

u(2 + 3u)

][
x− m(4 + 2u+ u2)

u(2 + 3u)

][
x− y +

2m(u− 2)

2 + 3u

]
,

ẏ =
[
y − m(3u2 − 2u− 4)

u(2 + 3u)

][
y − m(4 + 6u+ 3u2)

u(2 + 3u)

]
(2m− x+ y).

Since mu(1 + u)(2 + 3u) 6= 0 we can apply the transformation

x1 =
u(2 + 3u)

8m(1 + u)
x+

4 + 6u− u2

8(1 + u)
,

y1 =
u(2 + 3u)

8m(1 + u)
y +

4 + 2u− 3u2

8(1 + u)
, t1 =

64m2(1 + u)2t

u2(2 + 3u)2

and we obtain the system

ẋ = x(x− 1)
[
(1 + u)y − (1 + u)x+ u

]
, ẏ = y(y − 1)(u− x+ y).

It remains to observe that replacing in system (3.162) r = −1−u we obtain exactly
the above system, i.e. no new configurations could be obtained.



96 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Case (2r + 2 + u)(r + 1 − u) = 0. We observe that the condition (2r + 2 + u)2 +
(r+ 1− u)2 6= 0 has to be fulfilled, otherwise we obtain a contradiction: r+ 1 = 0.
So we examine both subcases given by this relation.

1. Subcase 2r + 2 + u = 0. Then r = −(u + 2)/2 6= 0 and we have r(r + 1) =
u(2 + u)/4 6= 0.

According to Lemma 3.17 in this case we could not have a couple of parallel
invariant lines in the direction y = −rx. Hence we deduce that the possible direc-
tions in which systems (3.142) could possess two couples of parallel invariant lines
could be either (i) x = 0 and y = 0, or (ii) x = 0 and y = x, or (iii) y = 0 and
y = x. We examine each one of these possibilities.

1.1. Directions x = 0 and y = 0. The existence of two couples of the parallel
invariant straight lines in these directions was examined earlier (see Lemma 3.18)
in the generic case and we have determined the conditions (3.145) (for direction
x = 0) and (3.146) (for direction y = 0). These conditions were detected without
imposing the condition 2r+ 2 +u 6= 0. The only restriction was (1 +u)(r+u) 6= 0,
which now becomes (1+u)(u−2) 6= 0. So the conditions (3.145) and (3.146) remain
valid also in the case 2r + 2 + u = 0 and setting r = −(u + 2)/2 we arrive at the
conditions

d = −
2
[
c(1 + u)2 − 2h2(2 + u)

]
(1 + u)(2 + u)

, a =
2h
[
2h2(2 + u)− c(1 + u)2

]
(1 + u)3

,

k = l = 0, e =
16m2 + f(u− 2)2

2(u− 2)
,

r = −(u+ 2)/2 b = −
4m
[
16m2 + f(u− 2)2

]
(u− 2)3

.

(3.173)

We point out that these conditions guarantee the existence of two couples of parallel
invariant lines in the directions x = 0 and y = 0. We recall that in this case the
parameter u must satisfy the following condition

u(u+ 1)(u− 2)(u+ 2) 6= 0, (3.174)

Now we consider the other two directions provided that the conditions (3.173)
are satisfied.

(a) Direction y = x. Considering (3.143) it remains to examine the equations
Eq6 = 0, Eq9 = 0 and Eq10 = 0. When the conditions (3.173) are satisfied we
obtain Eq6 = −2(h −m) = 0 and this yields h = m. By Lemma 2.8 in order to
have a common solution of the equations Eq9 = 0 nd Eq10 = 0 with respect to W

the condition R
(0)
W (Eq9, Eq10) = 0 is necessary. We calculate

R
(0)
W (Eq9, Eq10) =

uU1U2

2(u− 2)6(1 + u)6(2 + u)

where

U1 =2c(u− 2)2(1 + u)2 + f(u− 2)2(1 + u)2(2 + u)

+ 9m2u(2 + u)(4 + u),

U2 =(u− 2)4(1 + u)4(2c+ 2f + cu)2 − 4m2(u− 2)2(1 + u)2(4 + u)

×
[
c(2 + u)(16 + 6u+ u3)− f(32 + 36u+ 12u2 − u3)

]
+ 4m4(4 + u)2(4 + 2u+ u2)(112 + 112u+ 24u2 − 2u3 + u4)

(3.175)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 97

and because u 6= 0 we necessarily must have U1U2 = 0. We will later examine this
condition together with the conditions of the existence of one invariant line in the
direction y = −rx.

(b) The direction y = −rx. In this case, considering the equations (3.144) and
the conditions (3.173) and h = m, we obtain Eq6 = −u(2m+ 3W )/2 = 0 and due
to u 6= 0 this yields W = −2m/3. Clearly that this value of W must be a common
solution of the equations Eq9 = 0 nd Eq10 = 0 from (3.144). For this value of W
we obtain

Eq9 =
U ′1

9(u2 − 4)(1 + u)
, Eq10 =

U ′2
27(u− 2)3(1 + u)3(2 + u)

where

U ′1 = 9u(u− 2)(1 + u)(2c+ 2f + cu)−−2m2(4 + u)(−32− 16u+ 7u2),

U ′2 =m
[
9c(u− 2)3(8 + 5u)− 18f(u− 2)2(1 + u)3(16 + 2u+ u2)

− 2m2(4 + u)(128 + 1744u+ 1880u2 + 452u3 − 20u4 + 35u5)
]
.

It is clear that in order to have one invariant line in the direction y = −rx the
condition U ′1 = U ′2 = 0 must hold. In addition we must also have an invariant line
in the direction y = x and hence the following condition must be satisfied

U1U2 = U ′1 = U ′2 = 0,

where U1 and U2 are given in (3.175).
Since by condition (3.174) we have u(u+ 1)(u− 2) 6= 0, condition U ′1 = U ′2 = 0

gives

c = −2m2(2 + u)(64 + 64u+ 51u2 − 23u3 + 7u4)

9u2(u− 2)2(1 + u)2
,

f = −m
2(256 + 256u+ 204u2 + 232u3 + 109u4)

9u2(u− 2)2(1 + u)2
.

This implies U1 = 0 and we arrive at the family of systems

ẋ =
[
x+

2m(u− 8)

3u(u− 2)

][
x+

2m(8 + u+ 2u2)

3u(u− 2)(1 + u)

][
(1 + u)y − (2 + u)x/2

+
m(2 + u)

1 + u

]
,

ẏ =
[
y +

m(8 + 5u)

3u(1 + u)

][
y +

m(16 + 14u+ 7u2)

3u(u− 2)(1 + u)

][
(u− 2)x/2 + y − 4m

u− 2

]
.

(3.176)

To simplify these systems we need to use a transformation which depends on two
cases: m(4 + u) 6= 0 and m(4 + u) = 0.

1.1.1. Case m(4 + u) 6= 0. Then we can apply the transformation

x1 = −3u(u− 2)(1 + u)

2m(4 + u)2
x− (u− 8)(1 + u)

(4 + u)2
, t1 =

4m2(4 + u)4 t

9u2(u− 2)2(1 + u)2

y1 = −3u(u− 2)(1 + u)

2m(4 + u)2
y − (u− 2)(8 + 5u)

2(4 + u)2

and this leads to the 1-parameter family of systems

ẋ =x(x− 1)
[
2(1 + u)y − (2 + u)x+ u

]
/2,

ẏ =y(y − 1)
[
(u− 2)x+ 2y + u

]
/2.

(3.177)



98 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

It remains to observe that this family of systems is a subfamily of (3.151) defined
by the condition r = −(u+ 2)/2. So we could not obtain new configurations.

1.1.2. Case m(4 + u) = 0. If m = 0 then system (3.176) becomes cubic
homogeneous

ẋ = −x2(2x+ ux− 2y − 2uy)/2, ẏ = y2(−2x+ ux+ 2y) (3.178)

for which we calculate (see the definition of the polynomial H(X,Y, Z) on the page
13):

H(X,Y, Z) = gcd(G1,G2,G3) = X2(2X + uX − 2Y )(X − Y )2Y 2/16.

So by Lemmas 2.5 and 2.6 we conclude that the above system has invariant lines
of total multiplicity 8 (including the line at infinity).

Assume now m 6= 0 and u = −4. Then system (3.176) becomes

ẋ = (m− 3x)2(2m+ 3x− 9y)/27, ẏ = (m− 3y)2(2m− 9x+ 3y)/27

for which we also calculate

H(X,Y, Z) = −3−8(X − Y )3(3X + 3Y − 2mZ)(3X −mZ)2(3Y −mZ)2.

So in this case we obtain a system possessing invariant lines of total multiplicity 9.

1.2. Directions x = 0 and y = x. According to Lemma 3.18 for the existence
of a couple of parallel invariant straight lines in the direction x = 0 the conditions
(3.145) are necessary and sufficient. For r = −(u+ 2)/2 these conditions become

1 + u 6= 0, k = 0, d =
2h2(2 + u)− c(1 + u)2

(1 + u)(2 + u)
,

a =
2h2(2 + u)− c(1 + u)2

]
(1 + u)3

, r = −(2 + u)/2.

(3.179)

We consider now the direction y = x and ask for the existence of a couple of parallel
invariant straight lines in this direction. Considering (3.143) for r = −(u+ 2)/2 we
obtain Eq6 = l− 2h− k + 2m = 0, i.e. l = 2h+ k − 2m. Then taking into account
also (3.179) we calculate

Eq9 =
(1 + u)(2f + cu+ fu) + (2 + u)(e− 4h2 + eu)

2(1 + u)(2 + u)
+ 2(h−m)W

+ (4 + u)W 2/2,

Eq10 =
(1 + u)2(b+ 2ch+ bu)− 4h3(2 + u)

(1 + u)3
+ (e− c)W + 2(h−m)W 2

+ (2 + u)W 3/2.

We observe that Eq9 is of degree two with respect to the parameter W if and only
if u + 4 6= 0. Therefore in order to have two common solutions of the equations
Eq9 = 0 and Eq10 = 0 (i.e. to have a couple of parallel invariant lines, which could
coincide, in the direction y = x ) the condition u+ 4 6= 0 must be fulfilled.

On the other hand since by (3.174) the condition 2+u 6= 0 holds, then by Lemma
2.8 for the existence of two common solutions in W of the equations Eq9 = 0 and
Eq10 = 0 the following conditions are necessary and sufficient:

R
(0)
W (Eq9, Eq10) = R

(1)
W (Eq9, Eq10) = 0.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 99

Calculations yield

R
(1)
W (Eq9, Eq10) =

(4 + u)

4

[
2e−(2c+f)(2+u)

]
+h2+8hm−4m2+h2u+

3h2

1 + u
= 0

and from (u+ 1)(u+ 4) 6= 0 we obtain

e =
(2c+ f)(2 + u)

2
+

2h2(2 + u)

1 + u
− 8(h−m)2

4 + u
. (3.180)

Then condition R
(0)
W (Eq9, Eq10) = 0 gives

b =
4f(h−m)

4 + u
+

2c
[
h(3u2 + 2u− 4)− 4m(1 + u)2

]
(1 + u)(2 + u)(4 + u)

− 4h3(2 + u)

(1 + u)3
+

16h2(h−m)

(1 + u)(4 + u)
− 64(h3 − 3h2m+ 3hm2 −m3)

(4 + u)3

(3.181)

and therefore,

Eq9 = Ψ(W ), Eq10 =

[
8(h−m) + (2 + u)(4 + u)W

]
(4 + u)2

Ψ(W ),

Ψ(W ) =
(4 + u)(2c+ 2f + 2cu+ fu)

2(2 + u)
− h2(u2 + 4u+ 12)

(u+ 1)(u+ 4)
+

4m(2h−m)

u+ 4

+ 2(h−m)W + (4 + u)W 2/2.

Therefore the equations Eq9 = 0 and Eq10 = 0 have two common solutions in W
given by Ψ(W ) = 0, real or complex, distinct or coinciding.

Thus, considering the conditions (3.179), (3.180) and (3.181) we ask for the
existence of two invariant line in each one of the directions y = 0 and y = −rx.

(a) Direction y = 0. Taking into account (3.147) we obtain Eq5 = 2(h−m) = 0
and hence we obtain h = m. Then we calculate

Eq8 =
(2 + u)(2c+ f − 4m2 + 2cu+ fu)

2(1 + u)
− 2mW + (u− 2)W 2/2,

Eq10 = −
2m
[
c(1 + u)2 − 2m2(2 + u)

]
(1 + u)3

− fW −W 3.

Since by (3.174) the condition u − 2 6= 0 holds, according to Lemma 2.8, for the
existence of a common solution in W of the equations Eq8 = 0 and Eq10 = 0, the
following conditions are necessary and sufficient:

R
(0)
W (Eq8, Eq10) =

V1V2
2(1 + u)6

= 0, V1 = 2c(1 + u)2 + f(1 + u)2 −m2(7 + 4u),

V2 = c2(1 + u)4(2 + u)3 + 4c((1 + u)2)
[
f(1 + u)2(2 + u)2

−m2u(4 + u)(2 + 4u+ u2)
]

+ 4(2 + u)
[
f2(1 + u)4

− fm2u(1 + u)2(4 + u) +m4u2(4 + u)2
]
.

Hence we have to examine the condition V1V2 = 0. We will later examine this
condition together with conditions of the existence of one invariant line in the
direction y = −rx.

(b) Direction y = −rx. In this case from (3.144) considering the conditions
(3.179), (3.180) and (3.181) we obtain Eq6 = −u(2m + 3W )/2 = 0 which gives



100 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

W = −2m/3. Substituting in Eq9 and Eq10 from (3.144) we obtain

Eq9 =
V ′1

9(1 + u)(2 + u)
, Eq10 =

V ′2
27(1 + u)3(2 + u)

.

where

V ′1 = 9(u− 2)u(1 + u)(2c+ 2f + cu)− 2m2(4 + u)(16 + 7u),

V ′2 = 9(1 + u)2(2 + u)
[
c(u− 2)− 2f(1 + u)

]
+ 2m2(4 + u)3

So considering (3.174) the condition V ′1 = V ′2 = 0 gives

c =
2m2(31 + 29u+ 7u2)

9(1 + u)2(2 + u)
, f =

m2(2 + 19u+ 8u2)

9(1 + u)2(2 + u)

and this implies V1 = 0. So we arrive at the family of systems

ẋ =
[
x+

2m

3(2 + u)

][
x+

2m(5 + 2u)

3(1 + u)(2 + u)

][
(1 + u)y − (2 + u)x/2 +

m(2 + u)

1 + u

]
,

ẏ =
[
y +

m

1 + u

][ 4m2(5 + 2u)

9(1 + u)(2 + u)
+

3m(2 + u)

2(1 + u)
x− m

1 + u
y +

u− 2

2
xy + y2

]
.

If m = 0 then the above systems become exactly systems (3.178) which have in-
variant lines of total multiplicity 9.

In the case m 6= 0 via the transformation

x1 = − (1 + u)

2m(4 + u)

[
3(2 + u)x+ 2m

]
, t1 =

4m2t(4 + u)2

9(1 + u)2(2 + u)2
,

y1 = −3(1 + u)(2 + u)

2m(4 + u)
(x− y) + 1/2

we arrive at the family of systems

ẋ = x(x− 1)
[
ux− 2(1 + u)y− 2− u

]
/2, ẏ = −y(y− 1)

[
(4 + u)x− 2y+ 2 + u

]
/2.

It remains to observe that these systems become systems (3.177) by the change of
the parameter: u→ −(u+ 2).

1.3. Directions y = 0 and y = x. We claim, that this case could be brought to
the case of the directions x = 0 and y = x which we investigated above (beginning
from the page 96).

Recall that we are in the subcase 2r + 2 + u = 0. This condition depends only
on the parameters r and u of homogeneous cubic parts of systems (3.142) with
s = 1 (we denote this family (3.142) with {s = 1}). So we consider the associated
homogeneous cubic systems

ẋ = rx3 + (1 + u)x2y, ẏ = (r + u)xy2 + y3, r(r + 1) 6= 0 (3.182)

and apply the change (x, y, t, r, u) 7→ (y, x, r1t1, 1/r1, u1/r1). Then we obtain the
systems

ẋ = r1x
3 + (1 + u1)x2y, ẏ = (r1 + u1)xy2 + y3

with r1 = 1/r and u1 = u/r. For these systems we have

r1(r1 + 1) = (1 + r)/r2 6= 0, 2r1 + 2 + u1 = (2r + 2 + u)/r = 0.

Therefore we have passed from the homogeneous systems (3.182) with the condition
2r+2+u = 0 to the above systems with the condition 2r1+2+u1 = 0, interchanging
the directions y = 0, x = 0.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 101

It remains to observe that all the coefficients of non-cubic parts of the whole
system (3.142) are free parameters, except for the coefficient in front of x2 (respec-
tively y2) in the first (respectively second) equation, which vanishes. However the
change x↔ y does not affect this property.

2. Subcase r + 1− u = 0. According to Lemma 3.17 in this case we could have
a couple of parallel invariant straight lines in the direction y = −rx. We claim
that via a rescaling (which replaces the direction y = −rx with y = x) system
(3.142){s=1} with the condition r + 1 − u = 0 could be brought to system of the
same form but with the condition 2r + 2 + u = 0.

Indeed assume that for system (3.142) the condition r + 1 − u = 0 is satisfied.
Then applying the rescaling (x, y, t) 7→ (−x/r, y, t) to system (3.142){s=1} we obtain
the system

ẋ = a1 + c1x+ d1y + 2h1xy + k1y
2 + r1x

3 + (1 + u1)x2y,

ẏ = b1 + e1x+ f1y + l1x
2 + 2m1xy + (r1 + u1)xy2 + y3, r(r + 1) 6= 0

where a1, b1, . . . ,m1 are new free parameters and r1 = 1/r and u1 = −(r+1+u)/r.
Then r = 1/r1, u = −(r1 + 1 + u1)/r1 and for these system we have

r(r + 1) = (1 + r1)/r21 6= 0, r + 1− u = (2r1 + 2 + u1)/r1 = 0.

This completes the proof of our claim and this means that it is not necessary to
investigate the case r+ 1−u = 0 because no new configurations could be obtained.

Since all the possible cases were investigated we conclude that the class CSL4s∞
7

of cubic systems possess at most 93 distinct configurations of invariant lines. In the
next subsection we prove that all of them are non-equivalent according to Definition
1.3.

3.6. Geometric invariants and the proof of the non-equivalence of the 93
configurations. In this subsection we complete the proof of the Main Theorem
by showing that all 93 configurations of invariant lines we constructed are non-
equivalent according to Definition 1.3.

Notation 3.23. Let us denote

CS =
{

(S) : (S) is a system (2.1) with gcd(P (x, y), Q(x, y)) = 1 and

max
(

deg(P (x, y)),deg(Q(x, y))
)

= 3
}

CSL =
{

(S) ∈ CS : (S) possesses at least one invariant affine line or

the line at infinity with multiplicity at least two
}
.

Notation 3.24. Let

P̃ (X,Y, Z) = p0(a)Z2 + p1(a, X, Y )Z + p2(a, X, Y );

Q̃(X,Y, Z) = q0(a)Z2 + q1(a, X, Y )Z + q2(a, X, Y );

C̃(X,Y, Z) = Y P̃ (X,Y, Z)−XQ̃(X,Y, Z);

σ(P,Q) = {w ∈ R2)| P (w) = Q(w) = 0};

DS(P̃ , Q̃) =
∑

w∈σ(P̃ ,Q̃)

Iw(P̃ , Q̃)w;



102 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

DS(C̃, Z) =
∑

w∈{Z=0}

Iw(C̃, Z)w if Z - C̃(X,Y, Z);

DS(P̃ , Q̃;Z) =
∑

w∈{Z=0}

Iw(P̃ , Q̃)w;

D̂S(P̃ , Q̃, Z) =
∑

w∈{Z=0}

(
Iw(C̃, Z), Iw(P̃ , Q̃)

)
w,

where Iw(F,G) is the intersection number (see [14]) of the curves defined by homo-
geneous polynomials F,G ∈ C[X,Y, Z] and deg(F ),deg(G) ≥ 1.

A complex projective line uX + vY + wZ = 0 is invariant for a system (S) if
either it coincides with Z = 0 or it is the projective completion of an invariant
affine line ux+ vy + w = 0.

Notation 3.25. Let (S) ∈ CSL. Let us denote

IL(S) =
{
l : l is a line in P2(C) that is invariant for (S)},

M(l) = the multiplicity of the invariant line l of (S).

In defining M(l) we assume, of course, that (S) has a finite number of invariant
lines.

Remark 3.26. We note that the line L∞ : Z = 0 is included in IL(S) for any
(S) ∈ CSL.

Assume we have a finite number of invariant lines. Let li : fi(x, y) = ax+by+c =
0, i = 1, . . . , k, be all the distinct invariant affine lines (real or complex) of a
system (S) ∈ CSL. Let Li : Fi(X,Y, Z) = aX + bY + cZ = 0 be the complex
projective completion of li. Let mi be the multiplicity of the line Li and let m be
the multiplicity of the line at infinity Z = 0.

Notation 3.27. Let

G :
∏
i

Fi(X,Y, Z)miZm = 0;

Sing G = {w ∈ G : w is a singular point of G};
ν(w) = the multiplicity of the point w, as a point of G.

DIL(S) =
∑

l∈IL(S)

M(l)l, (S) ∈ CSL.

Next we define the geometric invariants which will be used in this work.

NR = #{l ∈ IL(S) : l : aX + bY + cZ = 0, a, b, c ∈ R};
MIL = max{M(l) : l ∈ IL(S)};

Nss =the total number of real singular points of the system which are located

on the smooth part of the total curve;

M(l∞) = the multiplicity of the invariant line l∞ : Z = 0 of (S);

Mσ = the maximum multiplicity of a real affine singular point of a system.

Suppose that a system (2.1) possesses a finite number of invariant lines L1, . . . Lk,
including the line at infinity. Sometimes it is convenient to consider in our discussion
a number of these invariant lines say Li1 , . . . Lil of a system (S). We call marked



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 103

system (S) by the lines Li1 , . . . Lil the object denoted by (S,Li1 , . . . Lil) of the
system (S) in which we singled out the lines Li1 , . . . Lil . We shall consider invariants
attached to such marked system.

For any non-degenerate cubic polynomial system (S) ∈ CSL marked with an
invariant line L that has a finite number of singularities, we can define the following
divisor:

Definition 3.28. Let D(S,L) =
∑
i ν(wi)wi where wi’s are the singular points of

the system (S) situated on the line L, and ν(wi) is the multiplicity of the point wi,
considered as a point of the curve G.

All singularities of the curve G are also singular points of the system. Apart
from these points, on G we may also have singular points of the system which are
smooth points of the curve G and these points play a role in constructing geometric
invariants for the system which will help distinguishing such system. Clearly for
such a point w we have ν(w) = 1.

Definition 3.29. We call s-point of the curve G a singular point of the system
that is a smooth point of the curve G.

Remark 3.30. Looking at the configurations we obtained for the class CSL4s∞
7 we

see that the maximum number of s-points located on a line is two, and that they
are never adjacent. Furthermore, for every system (S) ∈ CSL4s∞

7 with a simple
line L containing an s-point, the number of real singularities of (S) on L is 4. On
the other hand the maximum number of distinct invariant affine lines over R each
having an s-point is four. The maximum number of s-points occurring in a system
(S) is 4 and this occurs only in three configurations (Config. 7.66, Config. 7.89 and
Config. 7.91).

Consider a marked system (S,L) with the property that L is simple and on L
we have an s-point w. We attach an invariant to such marked system as follows:
Consider the divisor D(S,L) and order the integers ν(wi) appearing in this divisor
as follows: we start with the minimum of the integers ν(wi) which in this case is
1 = ν(w). We have two possibilities: (i) on L w is the only s-point and (ii) on L
we have two s-points.

Consider case (i) and look at the two adjacent points on L of the point w and
denote them by p and q and consider ν(p) and ν(q). Suppose that ν(p) 6= ν(q). Take
the smaller one of the two, suppose it is ν(p). This defines an order on L, namely
from w to p. Then order the singular points on L considering this order from w to p.
This gives us the ordered sequence of integers Sd(S,L) = (1, ν(p), ν(r), ν(q)) where r
is the fourth singular point on L of the system. Suppose now that ν(p) = ν(q). Then
since (1, ν(p), ν(r), ν(q)) = (1, ν(q), ν(r), ν(p)) we define Sd(S,L) as the common
value of these two sequences.

Consider now the case (ii). Suppose we have two s-points on L (= P1(R) ≡ S1),
call them w and p. According to Remark 3.30 these are not adjacent points. We
have 4 singular points on L which are w, r, p, q. Suppose that ν(r) < ν(q). We
define then Sd(S,L) = (1, ν(r), 1, ν(q)).

Clearly Sd is an invariant of marked system (S,L) corresponding to D(S,L).
We use the notation Sd to indicate that this is a sequence of integers attached to a
divisor, namely the divisor D(S,L).

Assume now that we have a number of straight invariant lines Li over R on
which we have affine s-points. Then for each such line we can form D(S,Li) and



104 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Figure 3. Diagram of non-equivalent configurations (to be continued)

Sd(S,Li). We now consider the marked system (S,Li; i ≤ 4) (see Remark 3.30)



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 105

Figure 3. (cont.) Diagram of non-equivalent configurations (to
be continued)

and to this we associate an invariant Ŝd = Sd(S,Li; i ≤ 4) defined as follows: We
first consider Sd(S,Li) for each one of the lines Li. We then define an order on
these Sd(S,Li), all of which start with 1. Consider two such 4-tuples Sd(S,Lj) and
Sd(S,Lk) and order these two 4-tuples according to the order of their corresponding

first elements where they do not coincide. We now define Ŝd = Sd(S,Li; i ≤ 4) as
this ordered sequence of these 4-tuples.

Let us consider an example: Config. 7.68, with three invariant affine lines with
s-points. Then the 4-tuples for the three invariant lines are: (1, 2, 3, 4), (1, 3, 2, 4)



106 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

Figure 3. (cont.) Diagram of non-equivalent configurations

and (1, 2, 4, 3). We now form the ordered 3-tuple of the 4-tuples: Ŝd =
(
(1, 2, 3, 4),

(1, 2, 4, 3), (1, 3, 2, 4)
)
.

Suppose the line at infinity L : Z = 0 is simple (i.e. of multiplicity one). Consider
the divisor D(S,L) and define S∞d (S,L) for the marked system (S,L) as follows:



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 107

On the line L we have four real singular points p, q, r, v of the system with cor-
responding multiplicities ν(p), ν(q), ν(r), ν(v) of these points considered as points
of G. Take the points with minimal multiplicity.

Consider first that we have a single point with minimal multiplicity. Without
loss of generality we may assume that this point is p. Consider the two adjacent
points of p which without loss of generality we may assume to be q, r and with their
multiplicities ν(q) ≤ ν(r). Consider the ordered 4-tuple (ν(p), ν(q), ν(v), ν(r)) and
define S∞d (S,L) as this 4-tuple. For example for the configuration Config. 7.26 we
have S∞d (S,L) = (1, 2, 3, 4).

Suppose now that we have two adjacent points p, q with the minimum multiplicity
as points of G. Consider the two remaining points r, v and we can assume without
loss of generality that ν(r) ≤ ν(v). We define S∞d (S,L) = (ν(p), ν(q), ν(r), ν(v)).

For example for the configuration Config.7.69 we have S∞d (S,L) = (2, 2, 3, 3).
Consider now the case when we have two singular points p, q of (S) with minimum

multiplicity i.e. ν(p) = ν(q), and which are not adjacent. Let the other two singular
points of (S) on L be r, v. Without loss of generality we may assume that ν(r) ≤
ν(v). Then for each one of the two points p, q we associate an order on L, i.e.
the order from p to r or the order from q to r. Consider the first order and
form the ordered 4-tuple of the multiplicities of the singular points listed in the
first order. We obtain (ν(p), ν(r), ν(q), ν(v)). For the second order we obtain
(ν(q), ν(r), ν(p), ν(v)). But these two 4-tuples are equal and we define S∞d (S,L) as
the common value of these two 4-tuples.

Consider now the case when we have at least three singular points p, q, r of
(S) on L with minimal multiplicity m ≤ ν(v) as points on G. Since these points
are on the line at infinity L then we necessarily must have three singular point
at infinity, all with multiplicity m, which are consecutive. In this case we define
S∞d (S,L) = (m,m,m, ν(v)).

Definition 3.31. Let (S) be a system (2.1) and suppose that at least one of the
coefficients jl in DL(S;Z) is three and that we actually have three parallel affine
lines, one of them L with real coefficients and two others complex: L′, L′′. Then as
indicated in the Notation 1.11 we can bring the cubic system to the form

ẋ = x[(x+ b)2 + u2], ẏ = q(a, x, y). (3.183)

which has the triplet of invariant lines: x = 0, x = −b+ iu, x = −b− iu. We have
here two possibilities: either b = 0 or b 6= 0. We define the invariant T R

L attached
to the marked system (S,L, L′, L′′) as follows: T R

L = 1 if and only if b 6= 0, and
T R
L = 0 if and only if b = 0.

Applying this invariant to configurations Config. 7.5 and Config. 7.6 we then
have in the first case T R

L (S) = 1 and in the second case T R
L (S) = 0.

The above definition is given for one triplet of invariant lines. In case we have two
such triplets (S,L0, L

′
0, L
′′
0), (S,L1, L

′
1, L
′′
1) we shall condense the notation by writ-

ing for T R
L the two corresponding coordinates which could be either (1, 1) or (1, 0)

or (0, 1) as defined in the previous definition. For example for the configuration
Config. 7.3 we have: T R

L = (1, 0).
Consider a system (S) and define MR

G(S) to be the maximum of the numbers
ν(p) where p is a real singular point of the curve G situated in the affine plane
and ν(p) is its multiplicity. Suppose that G possesses only one point w such that
ν(w) = MR

G . Consider all the invariant lines which pass trough w and consider



108 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

the points at infinity corresponding to all these lines. We now define an invariant
attached to such system (S): Define Adjw = k if and only if w is adjacent to exactly
k points at infinity. Then for example consider for the configuration Config. 7.68,
Adjw = 2 whereas for the Config. 7.71 we have Adjw = 3.

In a similar manner we introduce the invariant Adjs. Let (S) be a system of
our family. Consider all the invariant affine lines of (S) which have at least one
s-point. We define Adjs = 1 if and only if at least one of the s-points of these
lines is adjacent to the point at infinity of the line. Define Adjs = 0 if and only
if none of the s-points on these lines is adjacent to the point at infinity of the
corresponding line. Then for example consider for the configuration Config. 7.89,
Adjs = 1 whereas for the Config. 7.91 we have Adjs = 0.

We now introduce another invariant Jv as follows: Suppose a system (S) in our
family has a singular point of (S) v with maximum multiplicityMσ ≥ 2. We define
Jv to be 1 if and only if ν(v) =Mσ and 0 if and only if ν(v) 6=Mσ.

Acknowledgments. N. Vulpe is grateful for the hospitality of CRM Montreal
where this work was completed during his visit to CRM. D. Schlomiuk and N.
Vulpe were partially supported by the NSERC Grants RN000355 and RN001102.

References

[1] J. Artes, B. Grünbaum and J.Llibre; On the number of invariant straight lines for polynomial

differential system, Pac. J. Appl. Math., 184(1998), 317–327.
[2] V. A. Baltag; Algebraic equations with invariant coefficients in qualitative study of the

polynomial homogeneous differential system, Bul. Acad. Ştiinţe Repub. Mold. Mat., No. 42

(2003)(2), 31–46.
[3] V.A. Baltag, N.I. Vulpe; Total multiplicity of all finite critical points of the polynomial

differential system, Planar nonlinear dynamical system (Delft, 1995), Differ. Equ. Dyn.

Syst. 5 (1997), 455–471.
[4] C. Bujac; One new class of cubic system with maximum number of invariant lines omitted

in the classification of J. Llibre and N. Vulpe. Bul. Acad. Ştiinţe Repub. Mold., Mat., 75
(2014)(2), 102–105.

[5] C. Bujac; One subfamily of cubic system with invariant lines of total multiplicity eight

and with two distinct real infinite singularities, Bul. Acad. Ştiinţe Repub. Mold., Mat.,
77(2015)(1), 48–86.

[6] C. Bujac, N. Vulpe; Cubic differential system with invariant lines of total multiplicity eight

and with four distinct infinite singularities, J. Math. Anal. Appl., 423(2015), 1025–1080.
[7] C. Bujac, N. Vulpe; Cubic system with invariant straight lines of total multiplicity eight and

with three distinct infinite singularities, Qual. Theory Dyn. Syst., 14(2015)(1), 109–137.
[8] C. Bujac, N. Vulpe; Classification of cubic differential system with invariant straight lines

of total multiplicity eight and two distinct infinite singularities, Electron. J. Qual. Theory

Differ. Equ., 60(2015), 1–38.

[9] C. Bujac, N. Vulpe; Cubic Differential Systems with Invariant Straight Lines of Total Mul-
tiplicity Eight possessing One Infinite Singularity, Qual. Theory Dyn. Syst., 16(2017)(1),

1–30.
[10] C. Bujac, J. Llibre, N. Vulpe; First integrals and phase portraits of planar polynomial differ-

ential cubic system with the maximum number of invariant traight lines, Qual. Theory Dyn.

Syst., 15(2016)(2), 327–348.
[11] C. Christopher, J. Llibre, J. V. Pereira; Multiplicity of invariant algebraic curves in polyno-

mial vector fields, Pac. J. of Appl.Math., 329(2007)(1), 63–117.

[12] G. Darboux; Mémoire sur les équations différentielles du premier ordre et du premier degré,
Bulletin de Sciences Mathématiques, 2me série, 2 (1) (1878), 60-96; 123-144; 151-200.

[13] T. A. Druzhkova; Quadratic differential system with algebraic integrals, Qualitative theory

of differential equations, Gorky Universitet, 2 (1975), 34–42 (Russian).
[14] W. Fulton, W. A. Benjamin; Algebraic curves, Inc., New York, 1969, 226 pp.



EJDE-2021/?? CONFIGURATIONS OF INVARIANT LINES OF CUBIC SYSTEMS 109

[15] J. H. Grace, A. Young; The algebra of invariants, New York: Stechert, 1941.

[16] R. Hartshorne; Albegraic geometry, Graduate Texts in Math. 52, Springer, 1977.

[17] A. S. Householder; Bigradients and the problem of Routh and Hurwitz, SIAM Review, 10
(1968), 166–178.

[18] R. Kooij; Cubic system with four line invariants, including complex conjugated lines, Math.

Proc. Camb. Phil. Soc., 118 (1995)(1), 7–19.
[19] J. Llibre, D. Schlomiuk; Geometry of quadratic differential system with a weak focus of third

order, Canad. J. of Math., 56 (2004)(2), 310–343.

[20] J. Llibre, N.I. Vulpe; Planar cubic polynomial differential system with the maximum number
of invariant straight lines, Rocky Mountain J. Math., 38 (2006), 1301–1373.

[21] R. A. Lyubimova; On some differential equation possesses invariant lines, Differential and

integral eequations, Gorky Universitet 1 (1977), 19–22 (Russian).
[22] R. A. Lyubimova; On some differential equation possesses invariant lines, Differential and

integral equations, Gorky Universitet 8 (1984), 66–69 (Russian).
[23] P.J. Olver; Classical Invariant Theory, (London Mathematical Society student texts: 44),

Cambridge University Press, 1999.

[24] J. Pal, D. Schlomiuk; Summing up the dynamics of quadratic Hamiltonian system with a
center, Canad. J. Math., 49 (1997)(3), 583–599.

[25] M. N. Popa; The number of comitants that are involved in determining the number of integral

lines of a cubic differential system, Izv. Akad. Nauk Moldav. SSSR, Mat., 1 (1990), 67–69
(Russian).

[26] M. N. Popa, K. S. Sibirskii; Integral line of a general quadratic differential system, Izv. Akad.

Nauk Moldav.SSR, Mat., 1 (1991), 77–80 (Russian).
[27] M. N. Popa, K. S. Sibirskii; Conditions for the presence of a nonhomogeneous linear partial

integral in a quadratic differential system, Izv. Akad. Nauk Respub. Moldova, Mat., 3(1991),

58–66 (Russian).
[28] V. Puţuntică, A. Şubă; The cubic differential system with six real invariant straight lines along

three directions, Bull. of Acad. of Sci. of Moldova. Mathematics, 60(2009)(2), 111–130.
[29] V. Puţuntică, A. Şubă; Classiffcation of the cubic differential system with seven real invariant

straight lines, ROMAI J. 5(2009)(1), 121–122.

[30] D. Schlomiuk, J. Pal; On the geometry in the neighborhood of infinity of quadratic differential
phase portraits with a weak focus, Qual. Theory Dyn. Syst., 2 (2001)(1), 1–43.

[31] D. Schlomiuk, N. Vulpe; Planar quadratic differential system with invariant straight lines of

at least five total multiplicity, Qual. Theory Dyn. Syst., 5(2004), 135–194.
[32] D. Schlomiuk, N. Vulpe; Integrals and phase portraits of planar quadratic differential system

with invariant lines of at least five total multiplicity, Rocky Mountain J. Math., 38(2008)(6),

1–60
[33] D. Schlomiuk, N. Vulpe; Planar quadratic differential system with invariant straight lines of

total multiplicity four, Nonlinear Anal., 68(2008)(4), 681–715.

[34] D. Schlomiuk, N. Vulpe; Integrals and phase portraits of planar quadratic differential system
with invariant lines of total multiplicity four, Bull. of Acad. of Sci. of Moldova. Mathematics,

56(2008)(1), 27–83.
[35] D. Schlomiuk, N. Vulpe; Global classification of the planar Lotka–Volterra differential sys-

tem according to their configurations of invariant straight lines , J. Fixed Point Theory

Appl.,8(2010)(1), 69 pp.
[36] D. Schlomiuk, N. Vulpe; Global topological classification of Lotka-Volterra quadratic differ-

ential system, Electr. J. differ. equ., 2012 no. 64 (2012), 1–69.
[37] K. S. Sibirskii; Introduction to the algebraic theory of invariants of differential equations,

Nonlinear Anal Theory Methods Appl., Manchester University Press, Manchester, translated

from the Russian, 1988.

[38] J. Sokulski; On the number of invariant lines for polynomial vector fields, Nonlinearity, 9
(1996), 479–485.

[39] A. Şubă, V. Repeşco, V. Puţuntică; Cubic system with seven invariant straight lines of
configuration (3, 3, 1), Bull. of Acad. of Sci. of Moldova. Mathematics, 69(2012)(2), 81–98.

[40] A. Şubă, V. Repeşco, V. Puţuntică; Cubic system with invariant affine straight lines of total

parallel multiplicity seven, Electron. J. Diff. Eqns., Vol. 2013 (2013) no. 274, 1-22.

[41] N. I. Vulpe; Polynomial bases of comitants of differential system and their applications in
qualitative theory. (Russian) “Shtiintsa”, Kishinev, 1986.



110 C. BUJAC, D. SCHLOMIUK, N. VULPE EJDE-2021/??

[42] Zhang Xiang; Number of integral lines of polynomial system of degree three and four, J. of

Nanjing University, Math. Biquartely, 10 (1993), 209–212.

Cristina Bujac

Vladimir Andrunachievici Institute of Mathematics and Computer Science, Chişinău,
MD-2028, Moldova

Email address: crisulicica@yahoo.com

Dana Schlomiuk

Département de Mathématiques et de Statistiques, Université de Montréal, C.P. 6128,

succursale Centre-Ville, Montréal, (Québec) H3C 3J7, Canada
Email address: dana.schlomiuk@umontreal.ca

Nicolae Vulpe
Vladimir Andrunachievici Institute of Mathematics and Computer Science, Chişinău,

MD-2028, Moldova

Email address: nvulpe@gmail.com


	1. Introduction and statement of main results
	2. Preliminaries
	3. Proof of the main theorem
	3.1. Systems with the type of the configuration T=(3,3)
	3.2. Systems with configuration type T=(3, 2, 1)
	3.3. Systems with the configuration of the type T=(3,1,1,1)
	3.4. Systems with configuration type T=(2,2,2)
	3.5. Systems with configuration type T=(2,2,1,1)
	3.6. Geometric invariants and the proof of the non-equivalence of the 93 configurations
	Acknowledgments

	References