Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 69, pp. 1–52.

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

QUADRATIC SYSTEMS WITH AN INVARIANT ALGEBRAIC

CURVE OF DEGREE 3 AND A DARBOUX INVARIANT

JAUME LLIBRE, REGILENE D. S. OLIVEIRA, CAMILA A. B. RODRIGUES

Abstract. Let QS be the class of non-degenerate planar quadratic differential

systems and QS3 its subclass formed by the systems possessing an invariant

cubic f(x, y) = 0. In this article, using the action of the group of real affine
transformations and time rescaling on QS, we obtain all the possible normal

forms for the quadratic systems in QS3. Working with these normal forms we

complete the characterization of the phase portraits in QS3 having a Darboux
invariant of the form f(x, y)est, with s ∈ R.

1. Introduction and statements of results

Even after hundreds of studies on the topology of real planar quadratic vector
fields the complete characterization of their phase portraits is a quite complex task.
This family of systems depends on twelve parameters but, after affine transforma-
tions and time rescaling, we arrive at families with five parameters, which is still a
big number of parameters. Many subclasses have been considered.

Denote by R[x, y] the ring of the real polynomials in the variables x and y.
Consider the differential system in R2 given by

ẋ = P (x, y), ẏ = Q(x, y), (1.1)

where P,Q ∈ R[x, y]. Here the dot denotes derivative with respect to the time t
and the degree of system (1.1) is m = max{degP,degQ}.

When m = 2 we say that system (1.1) is a quadratic polynomial differential
system or simply a quadratic system. More than one thousand papers have been
published about quadratic systems, see for instance [15] for a bibliographical sur-
vey. The quadratic systems appear in the modeling of many natural phenomena
described in different branches of science, in biological and physical applications.
Besides the applications the quadratic systems became a matter of interest for the
mathematicians. Considering algebraic invariant curves, some authors have pub-
lished on the subject, see for instance [3] and [11]. In the first one the authors
studied cubic systems with invariant straight lines of total multiplicity eight that
have three distinct infinite singularities. The second paper is dedicated to study the
normal forms and global phase portraits of quadratic and cubic integrable systems
when they have two nonconcentric circles as invariant algebraic curves. The global

2010 Mathematics Subject Classification. 34C05, 34A34, 34C23.
Key words and phrases. Quadratic vector fields; algebraic invariant curve; Darboux invariant;

global phase portrait.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted September 25, 2020. Published August 16, 2021.

1



2 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

phase portrait of a quadratic system is also investigated in [13], where the authors
classified all global phase portraits of Bernoulli quadratic polynomial differential
systems in R2.

In this paper we assume that the polynomials P and Q are coprime, otherwise
system (1.1) can be reduced to a linear or constant system doing a rescaling of the
time variable.

The first objective of this paper is to characterize all quadratic systems having
invariant cubics. Then using the normal forms obtained, we investigate which
systems have a Darboux invariant either of the form estfλ1

1 fλ2
2 fλ3

3 if the cubic is

the product of three straight lines fi = 0 for i = 1, 2, 3 or of the form estfλ1
1 fλ2

2 if
the cubic is the product of one straight line f1 = 0 and an irreducible conic f2 = 0
or of the form estfλ1

1 if f1 = 0 is an irreducible cubic.
The paper is organized as follows. In section 2 we present our main results. They

are divided in two subsections. In section 3 we present definitions and results that
are used for proving our main results. Finally in sections 5, 6 and 7 we prove the
main results.

2. Statement of the main results

Since the cubic curves can be classified as reducible and irreducible curves (ac-
cording to the polynomial defining the curve admits fatorization or not), we split
the obtained results in two subsections. In the first one we consider planar quadratic
systems having irreducible cubics and in the second one, the reducible ones.

Theorem 2.1. Each quadratic system admitting an irreducible invariant cubic
after an affine change of coordinates and a rescaling of the time variable can be
written as one of the following systems, where a, b, c, d and r are real numbers,

(i) ẋ = 2(ax+ by + dxy + cx2),
ẏ = 3(ay + bx2 + cxy + dy2),

(ii) ẋ = 2(ax+ by + (3b− 2c)xy + ax2),
ẏ = 2bx+ 2ay + 2cx2 + 3axy + (9b− 6c)y2,

(iii) ẋ = 2(ax− by + (3b+ 2c)xy − ax2),
ẏ = 2bx+ 2ay + 2cx2 − 3axy + (9b+ 6c)y2,

(iv) ẋ = 2y(a+ bx),
ẏ = ar − 2(ar + a+ br)x+ (3a+ br + b)x2 + 3by2,

(v) ẋ = 2y(b+ cx),
ẏ = b+ 2(br − c)x+ (3b− cr)x2 + 3cy2.

Theorem 2.2. Each quadratic system admitting an irreducible invariant cubic
having a Darboux invariant can be written after an affine change of coordinates
and a rescaling of the time variable as

ẋ = x+ y, ẏ =
3

2
y + x2. (2.1)

After the change of coordinates, y2 − (2/3)x3 is the invariant algebraic curve and
the Darboux invariant is given by I1(t, x, y) = e−3t

(
2
3x

3 + y2
)
. The global phase

portrait of such system is given in Figure 1.

Theorems 2.1 and 2.2 are proved in section 5.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 3

Figure 1. Phase portrait of system (2.1).

2.1. Reducible invariant cubics. Each reducible cubic can be written as the
product of two polynomials one of degree two and the other of degree one. The con-
ics can be classified in ellipses (E), complex ellipses (CE), hyperbolas (H), parabolas
(P), two real straight lines intersecting in a point, two real parallel straight lines
(PL), one double invariant real straight line (DL), two complex straight lines in-
tersecting in a real point (p), and two complex parallel straight lines (CL). So the
normal forms of the reducible cubics, except to an affine transformation, are

(E) (x2 + y2 − 1)(ax+ by + c) = 0,
(CE) (x2 + y2 + 1)(ax+ by + c) = 0,

(H) (x2 − y2 − 1)(ax+ by + c) = 0,
(P) (y − x2)(ax+ by + c) = 0,

(LV) xy(ax+ by + c) = 0,
(PL) (x2 − 1)(ax+ by + c) = 0,
(DL) x2(ax+ by + c) = 0,
(CL) (x2 + 1)(ax+ by + c) = 0,

(p) (x2 + y2)(ax+ by + c) = 0.

We shall say that a quadratic system is of type (E) if it has a real ellipse and
a straight line as invariant irreducible algebraic curves; of type (CE) if it has a
complex ellipse and a straight line as invariant irreducible algebraic curves, and
respectively with all the nine types of conics described above.

The first result of this paper classifies the quadratic systems having a reducible
invariant cubic.

Theorem 2.3. If a quadratic system (1.1) has a reducible invariant cubic then it
can be written, after an affine change of coordinates, into one of the following forms

(CE) ẋ = −(x2 + y2 + 1)− 2α1y(y + ax+ c),
ẏ = a(x2 + y2 + 1) + 2α1x(y + ax+ c),

(E.1) ẋ = −(x2 + y2 − 1)− 2α1y(y + ax+ c),
ẏ = a(x2 + y2 − 1) + 2α1x(y + ax+ c),

(E.2) ẋ = (β1/2)(x2 + y2 − 1)− y(β2y − α2x+ cβ2),
ẏ = (y + c)(α2y − β2cx+ α2), where α2(c+ 1) = 0,

(H.1) ẋ = (β1/2)(x2 − y2 − 1) + β2y(y + c),
ẏ = β2y(y + c),

(H.2) ẋ = (x+ c)(α2x+ γ2y + α2),
ẏ = −(γ1/2)(x2 − y2 − 1) + x(γ2x+ α2y + cγ2), where α2(c+ 1) = 0,

(H.3) ẋ = (A/2)(x2 − y2 − 1)− y(α− cβ + x(β − cα))− y2(γ − cα),
ẏ = (A/2)(x2 − y2 − 1)− x(α− cβ + βx+ y(γ − cα)) + cα(y2 + 1),
where c(γ + β) = 0,



4 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

(H.4) ẋ = (A/2)(x2 − y2 − 1) + y(aα− β
√
d+ x(aβ − α

√
d) + βy),

ẏ = (−Aa/2)(x2 − y2 − 1) + x(aα− β
√
d+ aβx+ βy)− α

√
d(y2 + 1),

where d = a2 − 1,
(H.5) ẋ = −(x2 − y2 − 1) + 2α1y(y + ax+ c),

ẏ = a(x2 − y2 − 1) + 2α1x(y + ax+ c), where c2 6= a2 − 1,
(P.1) ẋ = x(α2 + β2x+ γ2y),

ẏ = α1(y − x2) + 2α2x
2 + 2y(β2x+ γ2y),

(P.2) ẋ = −β1(y − x2) + y(β2 + γ2x) + (α2 + γ2c)x+ cβ2,
ẏ = 2(y + c)(α2 + β2x+ γ2y), with cα2 = 0,

(P.3) ẋ = −(y − x2)− α(y + ax+ c),
ẏ = a(y − x2)− 2αx(y + ax+ c), with c 6= a2/4,

(LV.1) ẋ = x(α+ ry + βx),
ẏ = y(α+ (r − q + β)y + qx),

(LV.2) ẋ = x(p+ qx+ ry),
ẏ = y(y + c), with c(c+ 1) = 0,

(LV.3) ẋ = −x(y + α(y + ax+ c)),
ẏ = y(ax+ β(y + ax+ c)), with a c 6= 0,

(RPL) ẋ = x2 − 1,
ẏ = y(α+ βx+ γy),

(DL) ẋ = x2,
ẏ = y(α+ βx+ γy),

(CPL) ẋ = x2 + 1,
ẏ = y(α+ βx+ γy),

(p.1) ẋ = (β/2)(x2 + y2)− β3y
2 + x(α3 + γ3y),

ẏ = y(α3 + β3x+ γ3y),
(p.2) ẋ = −(x2 + y2) + (βx− αy)(y + ax+ c),

ẏ = a(x2 + y2) + (βy + αx)(y + ax+ c), with c 6= 0,

where a, c, d, A, p, q, r, α, β, γ, α1, β1, γ1, α2, β2, γ2, α3, β3 and γ3 are the parameters
of the system.

Theorem 2.4. The global phase portrait in the Poincaré disc of each quadratic
differential system admitting a reducible invariant cubic f(x, y) = 0 and having a
Darboux invariant of the form e−stf(x, y) is topologically equivalent to one of the
phase portraits presented in Figures 2-7. Their normal forms according to Theorem
2.3 is labelled in the corresponding figure.

Theorem 2.5. Systems of type (CE), (E.1), (H.1), (H.5), (P.3) do not admit Dar-
boux invariants of the form e−stf(x, y).

3. Preliminary and basic results

The goal of this section is introduce some definitions and results which are used
in next sections for the study of the Darboux invariants and to obtain the global
phase portrait of the systems of Theorems 2.2 and 2.3.

3.1. Invariants. A nonconstant C1 function H : U = R, defined in the open and
dense set U ⊂ R2, is a first integral of system (1.1) on U if H(x(t), y(t)) is constant
for all of the values of t for which (x(t), y(t)) is a solution of system (1.1) contained



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 5

Figure 2. Phase portraits of systems of type (E) and (H) when
they have a Darboux invariant. Phase portraits EL.2.1 and EL.2.2
correspond to system (E.2); HL.2.1–HL.2.3 correspond to system
(H.2); HL.3.1–HL.3.9 correspond to system (H.3). The dashed
lines denote a curve filled of singular points.

in U . In other words, H is a first integral of system (1.1) if and only if

P
∂H

∂x
+Q

∂H

∂y
= 0, for all (x, y) ∈ U.

An invariant of system (1.1) on the open subset U of R2 is a nonconstant C1

function I in the variables x, y and t such that I(x(t), y(t), t) is constant on all
solution curves (x(t), y(t)) of system (1.1) contained in U , i.e.

∂I

∂x
P +

∂I

∂y
Q+

∂I

∂t
= 0, for all (x, y) ∈ U. (3.1)

On the other hand, given f ∈ C[x, y], we say that the curve f(x, y) = 0 is an
invariant algebraic curve of system (1.1) if there exists K ∈ C[x, y] such that

P
∂f

∂x
+Q

∂f

∂y
= Kf. (3.2)



6 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Figure 3. Phase portraits of systems of type (P ) when they have
a Darboux invariant. Phase portraits PL.1.1–PL.1.24 correspond
to system (P.1).



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 7

Figure 4. Phase portraits of systems of type (P ) when they have
a Darboux invariant. Phase portraits PL.1.25–PL.1.30 correspond
to system (P.1); PL.2.1–PL.2.11 correspond to system (P.2). The
dashed lines denote a curve filled of singular points.

The polynomial K is called the cofactor of the invariant algebraic curve f = 0.
When K = 0, f is a polynomial first integral. Note that if a real polynomial
differential system has a complex invariant algebraic curve then it has also its
conjugate. It is important to consider the complex invariant algebraic curves of the



8 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Figure 5. Phase portraits of systems of type (LV ) when they have
a Darboux invariant. Phase portraits LVL.1.1–LVL.1.6 correspond
to system (LV.1); LVL.2.1–LVL.2.17 correspond to system (LV.2).
The dashed lines denote a curve filled of singular points.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 9

Figure 6. Phase portraits of systems of type (RPL) and (DL)
when they have a Darboux invariant. Phase portraits RPL.1–
RPL.17 correspond to system (RPL); DL.1–DL.3 correspond to
system (DL). The dashed lines denote a curve filled of singular
points.

real systems because sometimes these force the real integrability of the system, for
more details see Chapter 8 of [9], or the subsection 3.2.



10 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Figure 7. Phase portraits of systems of type (CPL) and (p) when
they have a Darboux invariant. Phase portraits CPL.1–CPL.7
correspond to system (CPL); p.1.2 and p.1.2 correspond to system
(p.1); p.2.1 and p.2.2 correspond to system (p.2). The dashed lines
denote a curve filled of singular points.

Let f, g ∈ C[x, y] and assume that f and g are relatively prime in the ring C[x, y],
or that g = 1. Then the function exp(f/g) is called a exponential factor of system
(1.1) if for some polynomial L ∈ C[x, y] of degree at most m− 1 we have

P
∂ exp(f/g)

∂x
+Q

∂ exp(f/g)

∂y
= L exp(f/g).

As previously we say that L is the cofactor of the exponential factor exp (f/g). We
observe that in the definition of exponential factor exp(f/g) if f, g ∈ C[x, y] then
the exponential factor is a complex function. Again when we look for a complex
exponential factor of a real polynomial system we are thinking the real polynomial
system as a complex polynomial system.

3.2. Darboux invariants. An invariant I is called a Darboux invariant if it can
be written into the form

I(t, x, y) = fλ1
1 · · · fλpp Fµ1

1 · · ·Fµqq es t,

where fi = 0 are invariant algebraic curves of system (1.1) for i = 1, . . . p, and Fj
are exponential factors of system (1.1) for j = 1, . . . , q, λi, µj ∈ C and s ∈ R \ {0}.
Observe that, if among the invariant algebraic curves a complex conjugate pair



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 11

f = Re(f) + Im(f)i = 0 and f̄ = Re(f) − Im(f)i = 0 occurs, then the Darboux

invariant has a factor of the form fλf̄ λ̄, which is the real multi-valued function((
Re(f)

)2
+
(

Im(f)
)2)Re(λ)

e−2 Im(λ) arctan(Im(f)/Re(f)).

So, if system (1.1) is real then the Darboux invariant is also real, independently of
the fact of having complex invariant curves or complex exponential factors. The
next result is proved in [9, Proposition 8.4].

Proposition 3.1. Suppose that f ∈ C[x, y] and let f = fn1
1 . . . fnrr be its factoriza-

tion into irreducible factors over C[x, y]. Then for a polynomial differential system
(1.1), f = 0 is an invariant algebraic curve with cofactor kf if and only if fi = 0
is an invariant algebraic curve for each i = 1, . . . , r with cofactor kfi . Moreover
kf = n1kf1 + . . .+ nrkfr .

The next result, proved in [6], explain how to obtain a Darboux invariant using
algebraic invariant curves of a polynomial differential system.

Proposition 3.2. Suppose that a polynomial system (1.1) of degree m admits p
invariant algebraic curves fi = 0 with cofactors ki for i = 1, . . . , p, q exponential
factors exp(gj/hj) with cofactors Lj for j = 1, . . . , q, then, if there exist λi and
µj ∈ C not all zero such that

p∑
i=1

λiki +

q∑
j=1

µjLj = −s, (3.3)

for some s ∈ R\{0}, then substituting fλii by |fi|λi if λi ∈ R, the real (multi-valued)
function

fλ1
1 . . . fλpp

(
exp

( g1

h1

))µ1

. . .
(

exp
( gq
hq

))µq
est

is a Darboux invariant of system (1.1).

The search of first integrals is a classic tool in order to describe phase portraits
of a 2–dimensional differential system. As usual the phase portrait of a system
is the decomposition of the domain of definition of this system as union of all its
orbits. It is well known that the existence of a first integral or an a invariant for a
planar differential system allow to draw its phase portrait. Here we investigate the
existence of invariants of the form f(x, y)est, called Darboux invariants, see section
3.2 for details. Such invariants describe the asymptotic behavior of the solutions
of the system. Indeed let φp(t) be the solution of system (1.1) passing through
the point p ∈ R2, defined on its maximal interval (αp, ωp) such that φp(0) = p. If
ωp =∞ we define the ω-limit set of p as

ω(p) = {q ∈ R2 : ∃{tn} with tn =∞ and φp(tn) = q when n =∞}.
In the same way, if αp = −∞ we define the α-limit set of p as

α(p) = {q ∈ R2 : ∃{tn} with tn = −∞ and φp(tn) = q when n =∞}.
For more details on the ω– and α–limit sets see for instance [9, section 1.4].

The existence of a Darboux invariant of system (1.1) provides information about
the ω– and α–limit sets of all orbits of system (1.1). More precisely, we have the
following result, where the definitions of Poincaré compactification and Poincaré
disc are given in subsection 4. Its proof can be found in [14].



12 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Proposition 3.3. Let I(t, x, y) = f(x, y)est be a Darboux invariant of system (1.1).
Let p ∈ R2 and φp(t) be the solution of system (1.1) with maximal interval (αp, ωp)
such that φp(0) = p. Assume s > 0. Then if ωp =∞ we have that ω(p) is contained
in the closure of {f(x, y) = 0} inside the Poincaré disc, and if αp = −∞ we have
that α(p) is contained in S1, i.e. at infinity. When s < 0 we interchange the roles
of ω(p) and α(p) with respect to s > 0.

4. Poincaré compactification

Let X = P (x, y) ∂
∂x +Q(x, y) ∂∂y be the planar polynomial vector field of degree m

associated to the polynomial differential system (1.1). The Poincaré compactified
vector field π(X ) corresponding to X is an analytic vector field induced on S2 as
follows (for more details, see [9]).

Let S2 = {y = (y1, y2, y3) ∈ R3 : y2
1 + y2

2 + y2
3 = 1} and TyS2 be the tangent

plane to S2 at point y. We identify R2 with T(0,0,1)S2 and we consider the central

projection f : T(0,0,1)S2 → S2. The map f defines two copies of X on S2, one in the
southern hemisphere and the other in the northern hemisphere. Denote by X ′ the
vector field D(f ◦ X ) defined on S2 \ S1, where S1 = {y ∈ S2 : y3 = 0} is identified
with the infinity of R2.

For extending X ′ to a vector field on S2, including S1, X must satisfy convenient
conditions. Since the degree of X is m, π(X ) is the unique analytic extension of
ym−1

3 X ′ to S2. On S2 \ S1 there are two symmetric copies of X , and once we know
the behavior of π(X ) near S1, we know the behavior of X in a neighborhood of the
infinity. The Poincaré compactification has the property that S1 is invariant under
the flow of π(X ). The projection of the closed northern hemisphere of S2 on y3 = 0
under (y1, y2, y3) 7→ (y1, y2) is called the Poincaré disc, and its boundary is S1.

Two polynomial vector fields X and Y on R2 are topologically equivalent if there
exists a homeomorphism on S2 preserving the infinity S1 carrying orbits of the flow
induced by π(X ) into orbits of the flow induced by π(Y) preserving or not the
orientation of all the orbits.

As S2 is a differentiable manifold, in order to compute the explicit expression of
π(X ), we consider six local charts Ui = {y ∈ S2; yi > 0} and Vi = {y ∈ S2; yi < 0},
where i = 1, 2, 3, and the diffeomorphisms Fi : Ui → R2 and Gi : Vi → R2, for i =
1, 2, 3, which are the inverses of the central projections from the tangent planes at
the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0,−1, 0), (0, 0, 1) and (0, 0,−1), respectively.
We denote by z = (u, v) the value of Fi(y) and Gi(y), for any i = 1, 2, 3, therefore
z means different things depending on the local charts where we are working. So
after some computations π(X ) is given by:

vm∆(z)
(
Q
(1

v
,
u

v

)
− uP

(1

v
,
u

v

)
,−vP

(1

v
,
u

v

))
in U1,

vm∆(z)
(
P
(u
v
,

1

v

)
− uQ

(u
v
,

1

v

)
,−vQ

(u
v
,

1

v

))
in U2,

∆(z)(P (u, v), Q(u, v)) in U3,

where ∆(z) = (u2 + v2 + 1)−(m−1)/2. The expressions for Vi’s are the same as that
for Ui’s but multiplied by the factor (−1)m−1. In these coordinates v = 0 always
denote the points of the infinity S1.

4.1. Irreducible invariant cubics. The next results characterize all irreducible
cubics, their proofs can be found in [2].



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 13

Proposition 4.1 ([2, Theorem 8.3]). A cubic is non-singular and irreducible and
has a flex (a generalized inflection point) if and only if it can be transformed with
affine transformations into either

y2 = x(x− 1)(x− r) with r > 1,

or

y2 = x(x2 + sx+ 1) with − 2 < s < 2.

Proposition 4.2 ([2, Theorem 8.4]). A cubic is singular and irreducible if and
only if it can be transformed with affine transformations into one of the forms

y2 = x3, y2 = x2(x+ 1), y2 = x2(x− 1).

Moreover in [2] it is proved that every non-singular and irreducible curve has a
flex. So we have the complete characterization of the irreducible cubics.

4.2. Reducible invariant cubics.

Proposition 4.3. A real quadratic system having an invariant conic after an affine
change of coordinates can be written in one of the following forms

• real ellipse

ẋ = (A/2)(x2 + y2 − 1) + 2y(p+ qx+ ry),

ẏ = (B/2)(x2 + y2 − 1)− 2x(p+ qx+ ry),

• complex ellipse

ẋ = (A/2)(x2 + y2 + 1) + 2y(p+ qx+ ry),

ẏ = (B/2)(x2 + y2 + 1)− 2x(p+ qx+ ry),

• hyperbola

ẋ = (A/2)(x2 − y2 − 1)− 2y(p+ qx+ ry),

ẏ = −(B/2)(x2 − y2 − 1)− 2x(p+ qx+ ry),

• parabola

ẋ = A(y − x2)− (p+ qx+ ry),

ẏ = B(y − x2)− 2x(p+ qx+ ry),

• Lotka-Volterra

ẋ = x(p1 + q1x+ r1y), ẏ = y(p2 + q2x+ r2y),

• two parallel real lines

ẋ = x2 − 1, ẏ = Q(x, y),

• (double line

ẋ = x2, ẏ = Q(x, y),

• two parallel complex lines

ẋ = x2 + 1, ẏ = Q(x, y),

• two non-parallel complex lines

ẋ = (A/2)(x2 + y2) + (C/2)x+ 2y(p+ qx+ ry),

ẏ = (B/2)(x2 + y2) + (C/2)y − 2x(p+ qx+ ry).



14 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Here A,B,C, p, q, r, p1, p2, q1, q2, r1, r2 are real parameters and Q(x, y) denotes an
arbitrary polynomial of degree 2.

The proof of the above result can be found in [4], except to the parabola that is
proved in [12]. The next result is due to Christopher, Llibre, Pantazi, Zhang and
Zholadek, see [7, 5, 17]. An algebraic proof of it also can be found in [5].

Theorem 4.4. Let fi = 0 for i = 1, . . . , q be q irreducible algebraic curves in C2,
and let k =

∑q
i=1 deg fi. We assume

(i) there are no points at which fi and its first derivatives all vanish,
(ii) the highest order terms of fi have no repeated factors,
(iii) no more than two curves meet at any point in the finite plane and are not

tangent at these points,
(iv) no two curves have a common factor in their highest order terms. Then

any polynomial vector field X of degree m tangent to all fi = 0 is of the
form describe bellow.
(a) If m > k − 1 then X = Y

(∏q
i=1 fi

)
+
∑q
i=1

(∏q
j=1,j 6=i fj

)
Xfi , where

Xfi = (−∂fi/∂y, ∂fi/∂x) is a Hamiltonian vector field, the hi are
polynomials of degree ≤ m− k + 1 and Y is a polynomial vector field
of degree less than or equal m− k.

(b) If m = k − 1 then X =
∑q
i=1 αi

(∏q
j=1,j 6=i fj

)
Xfi , where αi ∈ C. In

this case a Darboux first integral exists.
(c) If m < k − 1 then X ≡ 0.

Theorem 4.5 ([5, Lemma 7]). Assume that f = 0 and g = 0 are different irre-
ducible invariant algebraic curves of system (1.1) of degree m, and that they satisfy
conditions (i) and (iii) of Theorem 4.4. If gcd(fx, fy) = 1 and gcd(gx, gy) = 1,
then system (1.1) has the normal form

ẋ = Afg − h1fyg − h2fgy ẏ = Bfg + h1fxg + h2fgx,

where A,B and hj are polynomials, for i = 1, 2.

5. Proof of Theorems 2.1 and 2.2

From now on assume that P (x, y) = a00 + a01y + a02y
2 + a10x+ a11xy + a20x

2

and Q(x, y) = b00 + b01y + b02y
2 + b10x+ b11xy + b20x

2.

Proof of Theorem 2.1. If a quadratic system (1.1) has a singular irreducible invari-
ant cubic f(x, y) = 0 by Proposition 4.2 the function f can be written either as
f(x, y) = y2−x3 or f(x, y) = y2−x2(x+ 1) or f(x, y) = y2−x2(x− 1). The curve
f(x, y) = y2 − x3 = 0 is an invariant cubic for system (1.1) if and only if equation
(3.2) is satisfied. The solution of this equation in terms of the parameters of the
system is a00 = a02 = b00 = b10 = 0, b01 = 3a10/2, b02 = 3a11/2, b11 = 3a20/2,
b20 = 3a01/2. So the cofactor of f is K = 3(a10 + a20x + a11y). Setting a10 = a,
a20 = b, a01 = c, a11 = d and a rescaling of the time we obtain system (i) of
Theorem 2.1.

When f(x, y) = y2− x2(x± 1) we obtain the normal forms given in (ii) and (iii)
of the theorem following similar steps.

Now if a quadratic system (1.1) has an invariant non-singular irreducible cubic
f(x, y) = 0 then by Proposition 4.1 we can write f(x, y) = y2 − x(x − 1)(x − r)
with r > 1 or f(x, y) = y2 − x(x2 + sx + 1) with −2 < s < 2. In the first case



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 15

solving equation (3.2) we obtain three solutions but fixing r > 1 only one solution
can hold a00 = a02 = a10 = a20 = b01 = b11 = 0, b00 = a01r/2, b02 = 3a11/2,
b10 = −a01(r+ 1)−a11r, b20 = (3a01 +a11r+a11)/2. It corresponds to system (iv)
of Theorem 2.1.

For f(x, y) = y2 − x(x2 + sx+ 1) we obtain only one solution corresponding to
system (v) of the theorem. �

Using the normal forms described in Theorem 2.1 we investigate when these
systems admit a Darboux invariant of the form estf(x, y).

Proof of Theorem 2.2. First of all, it is easy to see that the cofactor K of f in sys-
tems (ii)–(v) of Theorem 2.1 has no constant terms. Then equation (3.3) becomes
λK+s = 0 which never holds if s ∈ R\{0} and λ ∈ C\{0}. Therefore we conclude
that systems (ii)–(v) do not admit a Darboux invariant of the form estf(x, y).

Now considering system (i) of Theorem 2.1. f(x, y) = y2−x3 = 0 is an invariant
curve of system (i) with cofactor K = 6(a + cx + dy). In this case the solution of
(3.3) is c = 0, d = 0, s = −6aλ. Taking λ = −s/(6a) we obtain the system

ẋ = 2(ax+ by), ẏ = 3(ay + bx2),

with Darboux invariant I(t, x, y) = e−6at(y2 − x3).
The normal form described in Theorem 2.2 is obtained doing the following change

of coordinates and rescaling of the time

x =
2a2

3b2
X, y =

2a3

3b3
Y, t =

1

2a
T.

In this case, the Darboux invariant is written as I1(t, x, y) = e−3t
(

2
3x

3 + y2
)
.

Now it remains to study the phase portrait of system (2.1). This system has two
singular points, namely z1 = (0, 0) hyperbolic unstable node, and z2 = (3/2,−3/2)
a hyperbolic saddle. Applying the Poincaré compactification, in the local chart U1

the compactified system has no singular points. However in the local chart U2 the
origin (0, 0) is a nilpotent singularity. With the notation of Theorem 3.5 of [9] the
compactified system has F (u) = −u5− (3/2)u6 and G(u) = −4u2− (7/2)u3. Hence
the origin of U2 is a nilpotent stable node. By the previous statements it follows
that the phase portrait of system (2.1) is the one illustrated in Figure 1. �

6. Proof of Theorem 2.3

The proof is divided in nine parts, according to the type of the conic in the
reducible cubic.

6.1. Systems of type (E). If system (1.1) has an invariant cubic of the form
f(x, y) = f1(x, y)f2(x, y) with f1 = x2 + y2− 1 and f2 = ax+ by+ c, then applying
a rotation we can assume b = 1. Therefore it follows from Proposition 3.1 that fj
is an invariant curve with cofactor kj = αj + βjx+ γjy, j = 1, 2. Now we consider
two possibilities: a = 0 and a 6= 0.

If a = 0 then using equation (3.2) we have Q = k2f2 and P = (k1f1 −
2yk2f2)/(2x). As P is a polynomial the parameters of the system must satisfy
on the of following conditions

s1 = {c = −1, α1 = 0, γ1 = 2α2, γ2 = α2},
s2 = {c = 1, α1 = 0, γ1 = −2α2, γ2 = −α2},



16 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

s3 = {α1 = 0, γ1 = 0, γ2 = 0}.

Moreover the solutions s1 and s2 provide equivalent systems, and we can summarize
the solutions s1 and s3 writing the system

ẋ =(β1/2)(x2 + y2 − 1)− y(β2y − α2x+ cβ2),

ẏ =(y + c)(α2y + β2cx+ α2),
(6.1)

with α2(c+ 1) = 0. This is exactly system (E.2) of Theorem 2.3.
When a 6= 0 we check when the hypotheses of Theorem 4.4 are satisfied. Clearly

f1 and f2 satisfies (i), (ii) and (iv). Condition (iii) is not satisfied when c2 = a2 + 1
because the line f2 = 0 is tangent to the real ellipse f1 = 0. Indeed if the straight
line f2 = y + ax + c = 0 is tangent to the real ellipse f1 = x2 + y2 − 1 = 0 at
the point (x0, y0), then their gradients are parallel in such point, what means that

x0−ay0 = 0. Replacing y0 = x0/a in the ellipse we conclude that x0 = ±a/
√
a2 + 1.

From f2 = 0 we obtain c = ∓
√
a2 + 1. Therefore the condition for the tangency is

c2 = a2 + 1. By a rotation we obtain f2 = y− 1. Again we are in system (6.1) with
c = −1.

Now assuming c2 6= a2 + 1, by Theorem 4.4 the considered system is given by

ẋ = −α2(x2 + y2 − 1)− 2α1y(y + ax+ c),

ẏ = aα2(x2 + y2 − 1) + 2α1x(y + ax+ c),
(6.2)

where α1, α2 ∈ C and a, c ∈ R. As we are looking for a real system, then α1, α2 ∈ R,
and doing a rescaling of the time we can assume α2 = 1. Note that system (6.2) is
exactly system (E.1) of Theorem 2.3.

6.2. Systems of type (CE). In this case we can follow the same steps applied
previously. If system (1.1) has an invariant cubic of the form f = f1f2 with f1 =
x2 + y2 + 1 and f2 = ax + by + c we suppose, without loss of generality, b = 1.
Since the coefficients a, b and c are real numbers the straight line f2 = 0 cannot be
tangent to the complex ellipse f1 = 0. So we obtain

ẋ = −α2(x2 + y2 + 1)− 2α1y(y + ax+ c),

ẏ = aα2(x2 + y2 + 1) + 2α1x(y + ax+ c),
(6.3)

where α1, α2 ∈ C and a, c ∈ R. Applying a rescaling we have α2 = 1 in (6.3), and
we obtain the normal form for the systems of type (CE).

6.3. Systems of type (H). Let f1 = x2− y2− 1 and f2 = ax+ by+ c be two real
algebraic invariant curves of system (1.1), so a2 + b2 6= 0. Proceeding as before if
a = 0 we can assume b = 1 and the system can be written in the form

ẋ = (β1/2)(x2 − y2 − 1) + β2y(y + c), ẏ = β2y(y + c),

with β1β2 6= 0. This is system (H.1) of Theorem 2.3.
If a 6= 0 and b = 0 we take a = 1 and system (1.1) satisfies P = k2f2 and

2y Q = 2xP −k1f1, where kj = αj+βjx+γjy, for j = 1, 2. Since Q is a polynomial
in the parameters of the system it must satisfy one of the following conditions

s1 = {c = −1, α1 = 0, β1 = 2α2, β2 = α2},
s2 = {c = 1, α1 = 0, β1 = −2α2, β2 = −α2},
s3 = {α1 = 0, α2 = 0, β1 = 0, β2 = 0}.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 17

Applying the change of coordinates x = −X, y = Y we conclude that case s1 and
s2 provide equivalent systems. Moreover we can summarize solutions s1 and s3 in
the unique system

ẋ = (x+c)(α2x+γ2y+α2), ẏ = −(γ1/2)(x2−y2−1)+x(γ2x+α2y+cγ2), (6.4)

with α2(c+ 1) = 0. System (6.4) corresponds to system (H.2) of Theorem 2.3.
If ab 6= 0 we assume b = 1 and consider three cases, according to the conditions

of Theorem 4.4. Note that condition (i) of Theorem 4.4 holds because ∇f1(x, y) =
(2x,−2y) and ∇f2(x, y) = (a, 1), where ∇ indicates the gradient. Condition (ii)
also holds. However condition (iv) is not verified when a2 − 1 = 0. Indeed in this
case f1 = (x+y)(x−y)−1 and f2 = (y±x)+c. Condition (iii) does not hold when
c2 = a2 − 1 since the straight line f2 = y + ax+ c = 0 is tangent to the hyperbola.
The proof of this last statement can be done analogously as for the systems of type
(E). Hence when a2 − 1 = 0 or c2 = a2 − 1 Theorem 4.4 does not hold and we split
the study of systems of type (H) for ab 6= 0 in three cases: a2 − 1 = 0, c2 = a2 − 1
and (a2 − 1)(c2 − a2 + 1) 6= 0.

For the first two cases we apply Propositions 3.1 and 4.3 to conclude that f1 is
an algebraic invariant curve of a quadratic system (1.1) and it can be written as

ẋ =
A

2
(x2−y2−1)−2y(p+qx+ry), ẏ = −B

2
(x2−y2−1)−2x(p+qx+ry), (6.5)

where A,B, p, q, r ∈ R. Fixing the cofactor of f2 = 0 as k2 = α + βx + γy, where
α, β, γ ∈ R and using system (6.5) we solve (3.2). First considering a = −1 (the
case a = 1 is analogous except by a reflection) equation (3.2) has two possible
solutions

s1 = {B = −A, c = 0, p = α/2, q = β/2, r = γ/2},
s2 = {B = −A+ 2cα, p = (αc− β)/2, q = (β − cα)/2, r = −(β + cα)/2, γ = −β}.
Using the two above solutions we obtain the system

ẋ =(A/2)(x2 − y2 − 1)− y(α− cβ + x(β − cα) + y(γ − cα)),

ẏ =(A/2)(x2 − y2 − 1)− x(α− cβ + βx+ y(γ − cα)) + cα(y2 + 1),

with c(γ + β) = 0. This is system (H.3) of Theorem 2.3.
Now considering c2 = a2− 1 we investigate the conditions that must be satisfied

by the parameters of system (6.5) in order that f2 = y + ax ±
√
a2 − 1 be an

invariant curve. Without loss of generality we can assume c =
√
a2 − 1. Equation

(3.2) has one solution, namely

B = aA− 2α
√
d, p = (β

√
d− aα)/2, r = −β/2,

q = (α
√
d− aβ)/2, γ = aβ − α

√
d,

where d = a2 − 1. Replacing it in (6.5) we obtain

ẋ =
A

2
(x2 − y2 − 1) + y(aα− β

√
d+ x(aβ − α

√
d) + βy),

ẏ =
A

2
a(x2 − y2 − 1) + x(aα− β

√
d+ aβx+ βy)− α

√
d(y2 + 1),

where d = a2 − 1, and this systems corresponds to system (H.4) of Theorem 2.3.
Finally if (a2 − 1)(c2 − a2 + 1) 6= 0, applying Theorem 4.4 we obtain the system

ẋ =− α2(x2 − y2 − 1) + 2α1y(y + ax+ c),



18 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

ẏ =aα2(x2 − y2 − 1) + 2α1x(y + ax+ c),

which is system (H.5) of Theorem 2.3.

6.4. Systems of type (P). Let f = (y−x2)(ax+by+c) = 0 be an invariant cubic
of system (1.1). When b = 0 we can assume f = x(y− x2). Indeed if b = 0 we take
a = 1 and make the change of coordinates x = X− c, y = Y −2cX+ c2. Using that
f2 = x = 0 is an invariant straight line we have P = k2f2 with k2 = α2 +β2x+γ2y,
and a quadratic system (1.1) can be written as

ẋ = x(α2 + β2x+ γ2y), ẏ = α1(y − x2) + 2α2x
2 + 2y(β2x+ γ2y). (6.6)

If b 6= 0 and a = 0 we can take b = 1 and proceed as in systems of type (H) and
(E), then we obtain the system

ẋ = −β1(y−x2)+y(β2+γ2x)+(α2+γ2c)x+cβ2, ẏ = 2(y+c)(α2+β2x+γ2y), (6.7)

with cα2 = 0. Observe that when c = 0 the invariant line is y = 0 and when α2 = 0
it is y + c = 0.

If ab 6= 0 and f2 = y ± ax + a2/4, f2 = 0 is tangent to the parabola. In this
case we can assume f2 = y + ax + a2/4 (the other case is a reflection). Applying
the change of coordinates x = −X − a/2 and y = Y + aX + a2/4 the cubic
f = (y − x2)(y + ax + a2/4) becomes f = (Y − X2)Y , which already has been
studied above. Indeed it corresponds to system (6.7) with c = 0.

Otherwise there is no tangency between the straight line and the parabola, and
we apply Theorem 4.5 to get the differential system

ẋ = −(y − x2)− α(y + ax+ c), ẏ = a(y − x2)− 2αx(y + ax+ c). (6.8)

Systems (6.6), (6.7) and (6.8) correspond to systems (P.1), (P.2) and (P.3) of The-
orem 2.3, respectively.

6.5. Systems of type (LV). In this case f = xy(ax+ by+ c) = 0 is the invariant
curve and except by a rotation we can assume b = 1. We consider different cases
according to ac = 0 or ac 6= 0. Note that if c = 0 hypothesis (iii) of Theorem 4.4 is
not valid, whereas a = 0 breaks the hypothesis (iv).

When c = 0 and a 6= 0, doing the change of coordinates x = − Y
3√
a2

, y = 3
√
aX the

cubic becomes F = XY (Y −X). So using Proposition 4.3 the differential system
can be written as

ẋ = x(p1 + q1x+ r1y), ẏ = y(p2 + q2x+ r2y). (6.9)

If (6.9) has f3 = y − x as an invariant curve with cofactor k = α + βx + γy, then
equation (3.2) must be satisfied. Solving it we obtain

s1 = {p2 = α, r2 = β − q2 + r1, q1 = β, p1 = α, γ = β − q2 + r1}.

Replacing in (6.9) and writing q = q2, r = r1 we obtain system (LV.1).
Now if c = a = 0 then f2 = y = 0 is a double line, and it is not difficult to see

that we can write the system as

ẋ = x(p+ qx+ ry), ẏ = y2. (6.10)

Finally, when a = 0 and c 6= 0, doing the change of coordinates x = X/c2, y =
cY − c the cubic f = 0 becomes F = XY (Y − 1). So without loss of generality we
can work with f3 = y − 1. Again the idea is to write the system as in (6.9), and



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 19

see what are the conditions in order that f3 = 0 to be an invariant curve for such
system. Solving equation (3.2) and replacing the solutions in (6.9) we obtain

ẋ = x(p+ qx+ ry), ẏ = y(y − 1). (6.11)

Systems (6.10) and (6.11) can be summarized as

ẋ = x(p+ qx+ ry), ẏ = y(y + c),

with c = 0 or c = −1. This is exactly system (LV.2) of Theorem 2.3.
In the last case, a c 6= 0 the invariant cubic is f = xy (y + ax + c) = 0 and by

the geometry to the curves we can assume a < 0 and c < 0. Applying Theorem 4.4
we obtain the system

ẋ = −α2x(y + ax+ c)− α3xy, ẏ = α1y(y + ax+ c) + aα3xy.

Note that we can take α3 = 1. Doing α = α2, β = α1 we obtain system (LV.3).

6.6. Systems of type (RPL). Here the invariant cubic is f = f1f2f3 = 0 where
f1 = x + 1, f2 = x − 1 and f3 = ax + by + c. When b = 0 we apply Proposition
4.3 (case (RPL)), then it is easy to see that the corresponding normal form has
one additional invariant curve f3 = 0 as invariant straight line if and only if it is a
multiple of f1 or f2. However we cannot consider any of these cases because if the
system has f2 as an invariant double straight line for example, then there would be
a change of coordinates so that the system would be written as

ẋ = (x− 1)(x+ 1)2, ẏ = Q(x, y),

then having degree 3 instead of 2.
When b 6= 0 we can fix b = 1. In this case the cubic f = (x2− 1)(y+ ax+ c) = 0

can be reduced to F = y(x2− 1) by change of coordinates x = X, y = Y − aX − c.
If the quadratic differential system (1.1) has the invariant curve f = y(x2− 1) = 0,
then f1 = 0 and f2 = 0 are invariant curves and by Proposition 4.3 such system
can be written as

ẋ = x2 − 1, ẏ = Q(x, y),

where Q(x, y) is an arbitrary polynomial of degree 2. Imposing that f3 = y = 0
is an additional invariant curve with cofactor k3 = α+ βx+ γy, the above system
must satisfy Q(x, y) = y(α + βx + γy). This expression justify the normal form
given in (RPL) of Theorem 2.3.

6.7. Systems of type (DL). These systems have a double straight line as invari-
ant curve which can be taken as f1 = x. We write f2 = ax + by + c and use the
normal form of a system having f = f1f2 = 0 as an invariant cubic. For such
normal form, if b = 0 then f2 = 0 is an invariant straight line if and only if c = 0
but in this case the system cannot have a triple invariant straight line.

If b 6= 0 we can take b = 1 and f = x2(y + ax + c). Doing the change x = X,
y = Y − aX − c the function f can be written as F = X2Y . Hence it is enough to
consider f2 = y. By Proposition 4.3 a quadratic system (1.1) can be written as

ẋ = x2, ẏ = Q(x, y),

where Q(x, y) is an arbitrary polynomial of degree 2. Imposing that f2 = 0 is
an additional invariant curve with cofactor k2 = α + βx + γy, we conclude that
Q(x, y) = y(α+ βx+ γy). This expression justify the normal form given by (DL).



20 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

6.8. Systems of type (CPL). The proof for this case is analogous to the case
(DL) so we will omit some details. In short the cubic is given by f = f1f2f3 = 0
where f1 = x + i, f2 = x − i and f3 = ax + by + c. In order for f3 = 0 to be an
invariant curve with b = 0 it is necessary that c = ±i. So b 6= 0 and we assume
b = 1. This reduce f to the cubic F = y(x2 + 1) and then we obtain the normal
form (CPL) described in Theorem 2.3.

6.9. Systems of type (p). In this case the cubic is given by f = (x2 + y2)(ax+
by + c) = 0 and except by a rotation we can assume b = 1. When c = 0 the
three curves intersect at the same point and the conditions of Theorem 4.4 are not
satisfied. But if c = 0 doing the change of coordinates

x = − X
3
√

(a2 + 1)2
+

aY
3
√

(a2 + 1)2
, y =

aX
3
√

(a2 + 1)2
+

Y
3
√

(a2 + 1)2
,

the cubic f = (x2 +y2)(y+ax) = 0 is reduced to f = Y (X2 +Y 2). Now using that
system (1.1) has f3 = y = 0 as a third invariant curve it follows that Q(x, y) = k3f3

where k3 = α3 + β3x + γ3y is the cofactor of f3. Moreover f1f2 = 0 is also an
invariant curve then we must have

2xP (x, y) + 2yQ(x, y) = k(x, y)(x2 + y2),

with k(x, y) = α + βx + γy being the sum of the cofactors of f1 and f2. So a
quadratic system (1.1) can be written as

ẋ = (β/2)(x2 + y2)− β3y
2 + x(α3 + γ3y), ẏ = y(α3 + β3x+ γ3y),

which is exactly system (p.1) of Theorem 2.3.
When c 6= 0 we apply Theorem 4.4 and conclude that a quadratic system (1.1)

can be written as

ẋ = −α3(x2 + y2)− ((α2 + α1)y − i(α2 − α1)x)(y + ax+ c),

ẏ = aα3(x2 + y2) + ((α2 + α1)x− i(α2 − α1)y)(y + ax+ c),
(6.12)

with α1, α2 and α3 ∈ C. Writing αj = mj + i nj with mj , nj ∈ R and using
that such system have real parameters we conclude that m2 = m1, n2 = −n1 and
n3 = 0. Replacing this conditions in (6.12) we obtain the system

ẋ =−m3(x2 + y2) + 2(n1x−m1y)(y + ax+ c),

ẏ =am3(x2 + y2) + 2(m1x+ n1y)(y + ax+ c).

Note that if m3 = 0 then the system has a common factor, so we can take m3 = 2.
After a rescaling of the time and writing α = m1, β = n1 we obtain system (p.2).

It follows from the previous study the proof of Theorem 2.3.

7. Proof of Theorems 2.4 and 2.5

In this section we investigate planar quadratic systems with algebraic invariant
cubics having Darboux invariant. Moreover we investigate the phase portraits, in
the Poincaré disc, of such systems.

Proposition 7.1. Each real planar quadratic differential system with a real invari-
ant ellipse and an invariant straight line having a Darboux invariant can be writ-
ten, after an affine change of coordinates, as system (E.2) with c = −1, β1 = 2β2,
α2 6= 0. Moreover, such systems have the Darboux invariant of the form

I2(t, x, y) = e−t(y − 1)1/α2(x2 + y2 − 1)−
1

2α2 .



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 21

and, they have only two non equivalent phase portraits, see phase portraits EL.2.1
and EL.2.2 of Figure 2.

Proof. If follows from the reducible cubic classification that we can fix f1 = x2 +
y2 − 1 = 0 as the real ellipse and by Theorem 2.3 there are only two families of
systems having f1 = 0 and a straight line as invariant curves (E.1) and (E.2). We
shall prove later that (E.1) does not admit a Darboux invariant. Now we study
system (E.2). By Proposition 3.2 system (E.2) has a Darboux invariant if there
exist λ1, λ2 ∈ R not both equal to zero such that (3.3) holds with s ∈ R \ {0} and
k1, k2 being the cofactors of f1 = 0 and f2 = y+c = 0, respectively. But for system
(E.2) we must have α2 = 0 or c = −1. If α2 = 0 the cofactors are k1 = β1x and
k2 = β2x and the equation λ1k1 + λ2k2 + s = 0 has no solution for s 6= 0. Hence if
α2 = 0 system (E.2) has no Darboux invariant.

If α2 6= 0 and c = −1 then the cofactors are k1 = β1x + 2α2y and k2 =
α2 + β2x+ α2y and the unique solution of (3.3), with s 6= 0 is

β1 = 2β2, s = −α2λ2, λ1 = −λ2/2. (7.1)

Taking λ1 = 1/α2 and replacing (7.1) in system (E.2) we obtain system

ẋ = β2(y − 1) + x(β2x+ α2y), ẏ = (y − 1)(α2 + β2x+ α2y), (7.2)

which has the Darboux invariant

I2(x, y, t) = e−t(y − 1)1/α2(x2 + y2 − 1)−
1

2α2 .

To study the global phase portrait of system (E.2) we start considering its finite
singularities. Note that (7.2) has at most three finite singularities, namely z1 =

(0, 1), z2 = (−1/β2, 1) and z3 =
(
− 2β2

β2
2+1

,
β2
2−1

β2
2+1

)
. The eigenvalues associated to

z1 are 2 and 1, if β2 6= 0, the eigenvalues associated to z2 are −1 and 1 and the
eigenvalues of z3 are −1 and −2. So for β2 6= 0 z1, z2 and z3 are an unstable node,
a saddle and a stable node, respectively. When β2 = 0 we have only z1 and z3 as
finite singularities.

In the local chart U1 the compactified system is

u̇ = −v(β2 + β2u
2 − β2uv + v), v̇ = −v(β2 + β2uv + u− β2v

2), (7.3)

so v = 0 is a common factor, this means that v = 0 is a line of singular points.
Eliminating the common factor v, system (7.3) has no singular points if β2 6= 0.
Otherwise u1 = (0, 0) is a singular point with eigenvalues −1 and 1 which implies
that the origin is a hyperbolic saddle besides the line of singular points.

In the local chart U2 the compactified system is written as

u̇ = v(β2 + β2u
2 + uv − β2v), v̇ = v(v − 1)(β2u+ v + 1).

With a rescaling of the time the common factor v is eliminated and we can see that
(0, 0) is not a singular point of the compactified system.

Note that if β2 = 0 there are an additional invariant straight line given by
y + 1 = 0. From the study of the finite and infinite behavior of system (E.2) we
conclude that such system has two non-equivalent phase portraits when c = −1:
EL.2.1, if β2 6= 0 and EL.2.2, if β2 = 0. See Figure 2. �

Proposition 7.2. Each real planar quadratic differential system with an invariant
hyperbola and an invariant straight line having a Darboux invariant can be written,
after an affine change of coordinates, as



22 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

(i) system (H.2) with α2 6= 0, c = −1 and γ1 = 2γ2. Its Darboux invariant is

I3(x, y, t) = e−α2t(x2 − y2 − 1)−1/2(x− 1).

ii) system (H.3) with Aα 6= 0, c = 0 and β = −γ. Its Darboux invariant is

I4(x, y, t) = e−Aαt(x2 − y2 − 1)γ(y − x)A.

(iii) system (H.3) with α 6= 0 and β = γ = 0. Its Darboux invariant is

I5(x, y, t) = eαt(y − x+ c)−1.

(iv) system (H.4) with α 6= 0 and A = 2β. Its Darboux invariant is

I6(x, y, t) = e−αt(x2 − y2 − 1)−1/2(y + ax−
√
a2 − 1).

Moreover there are 12 non-equivalent phase portrait in the Poincaré disc of these
systems. They are in Figure 2 HL.2.1–HL.2.3, HL.3.1–HL.3.9.

Proof. Fixing f1 = x2 − y2 − 1 = 0, Proposition 3.2 says that system (H.2) has a
Darboux invariant if equation (3.3) holds for λ1, λ2 not both zero, where s ∈ R\{0},
and k1, k2 are cofactors of f1 = 0 and f2 = x+c = 0, respectively. Moreover c = −1
or α2 = 0 in system (H.2). For α2 = 0 we have k1 = γ1y and k2 = γ2y and the
equation λ1k1 + λ2k2 + s = 0 has no solution with s 6= 0. So in this case system
(H.2) has no Darboux invariant. If α 6= 0 and c = −1 then k1 = 2α2x + γ1y and
k2 = α2 + α2x + γ2y and (3.3) has a unique solution s = −α2λ2, γ1 = 2γ2, λ1 =
−λ2/2. The proof of (i) follows taking λ2 = 1 and replacing γ1 = 2γ2 in system
(H.2), so we obtain that system

ẋ = (x− 1)(α2 + α2x+ γ2y), ẏ = −γ2(x2 − y2 − 1) + x(−γ2 + γ2x+ α2y), (7.4)

has the Darboux invariant

I3(x, y, t) = e−α2t(x2 − y2 − 1)−1/2(x− 1).

To prove (ii) and (iii) we study system (H.3) considering two cases: c = 0 and
β = −γ. It is easy to see that if c = 0 (H.3) has a Darboux invariant when α 6= 0
and β = −γ. In this case we obtain

ẋ = (A/2)(x2−y2−1)−y(α−γx+γy), ẏ = (A/2)(x2−y2−1)−x(α−γx+γy),

with the Darboux invariant

I4(x, y, t) = e−Aαt(x2 − y2 − 1)γ(y − x)A.

If β = −γ, system (H.3) has a Darboux invariant only when γ = 0 and α 6= 0. In
this case, we obtain to system

ẋ = (A/2)(x2 − y2 − 1)− αy(1− cx− cy),

ẏ = (A/2)(x2 − y2 − 1)− αx(1− cy) + c α(y2 + 1),

with the Darboux invariant

I5(x, y, t) = eαt(y − x+ c)−1.

The study of (iv) follows from system (H.4) where the invariant line is f2 =

y + ax −
√
a2 − 1 = 0. In this case the unique solution of (3.3) is s = −αλ2,

A = 2β, λ1 = −λ2/2. So taking λ2 = 1 we obtain the Darboux invariant

I6(x, y, t) = e−αt(x2 − y2 − 1)−1/2(y + ax−
√
a2 − 1).



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 23

Now, let us study of possible phase portraits of system (7.4). Since α2 6= 0 we
can take α2 = 1 and the transformation x = X, y = −Y takes the system with
parameter γ2 to the system with parameter −γ2. So we may also assume γ2 ≥ 0.

Considering the finite singularities, if γ2 6∈ {0, 1} system (7.4) has three finite

singularities, namely z1 = (0, 1), z2 = (1,−1/γ2) and z3 =
(γ2

2+1

γ2
2−1

,− 2γ2
γ2
2−1

)
. The

eigenvalues associated to z1 are 2 and 1, if β2 6= 0, the eigenvalues associated to z2

are −1 and 1 and the eigenvalues of z3 are −1 and −2. So for γ2 6∈ {0, 1} z1, z2 and
z3 are respectively, an unstable node, a saddle and a stable node. When β2 = 0 we
have only z1 and z3 as finite singularities.

In the local chart U1 the compactified system is

u̇ = v(−γ2 + γ2u
2 + uv + γ2v), v̇ = v(v − 1)(γ2u+ v + 1), (7.5)

so v is a common factor, this means that v = 0 is a line of singular points. Eliminat-
ing the common factor v, system (7.5) has no singular points if γ2 6= 1. Otherwise
u1 = (−1, 0) is a singular point with eigenvalues −2 and −1, which implies that
u1 is a hyperbolic stable node. Moreover if γ2 = 0 there an additional invariant
straight line given by x+ 1 = 0.

In the local chart U2 the compactified system is written as

u̇ = −v(γ2 − γ2u
2 + γ2uv + v), v̇ = −v(γ2 + γ2v

2 − γ2uv + u).

So applying a rescaling of the time to eliminate the common factor v, we obtain
that the origin is a singular point of the compactified system if and only if γ2 = 0.
In this case (0, 0) is a hyperbolic saddle.

It is easy to see that if γ2 ∈ (0, 1) the singularities z1 and z3 are in distinct
branches of the hyperbola, and if γ2 ∈ (1,+∞) they are in the same branch as
shows Figure 8.

Figure 8. Possible phase portraits of sytem (7.4) when γ2 6∈ {0, 1}.

From [9, Theorem 1.43] (Markus-Neumann-Peixoto Theorem) we conclude that
these two phase portraits are topologically equivalent. By continuity and the study
done previously we conclude that system of type (H.2) having a Darboux invariant
can have three non-equivalent phase portrait. The case γ2 6= 0, 1 corresponds to
HL.2.1 in Figure 2 and when γ2 = 1 or γ2 = 0 we have the phase portraits HL.2.2
and HL.2.3 of Figure 2, respectivelly. �

Now we study the global phase portrait of system (H.3). Remember that the
parameters of (H.3) must satisfy c(γ + β) = 0. We start considering c = 0, then
the differential system is

ẋ = (A/2)(x2 − y2 − 1)− y(α− γx+ γy),

ẏ = (A/2)(x2 − y2 − 1)− x(α− γx+ γy),
(7.6)



24 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

which has f1 = x2 − y2 − 1 = 0 and f2 = y − x = 0 as invariant algebraic curves.
Since α 6= 0 we can take α = 1 and the transformation x = −X, y = −Y allows to
assume A > 0.

If γ 6= 0 then z1 = (−A/2,−A/2) and z2 =
(
(γ2 + 1)/(2γ), (γ2 − 1)/(2γ)

)
are

the two finite singular points. If γ = 0 exists only one finite singular point.
The eigenvalues associated to z1 are −1 and 1 so z1 is a saddle. The eigenvalues

associated to z2 are A/γ and −1, so z2 is a stable node if γ < 0, and a saddle if
γ > 0. Moreover z1 is on the straight line and z2 is on the hyperbola.

In the local chart U2, we have system

u̇ = (1/2)(u− 1)(Av2 − (A+ 2γ)u2 + 2uv + 2v +A+ 2γ),

v̇ = (1/2)v(Av2 − (A+ 2γ)u2 + 2γu+ 2uv +A),

and the origin is a singular point only when A + 2γ = 0 but in this case the line
v = 0 is filled up of singular points.

In the local chart U1, we have system

u̇ = (1/2)(u− 1)((A+ 2γ)u2 +Av2 + 2uv + 2v −A− 2γ),

v̇ = (1/2)v((A+ 2γ)u2 +Av2 + 2uv − 2γu−A),

which has the infinity filled up by singularities when A + 2γ = 0, otherwise, there
are two singularities u1 = (−1, 0) and u2 = (1, 0).

Assuming A+ 2γ 6= 0. The point u1 has eigenvalues 2γ and 2(A+ 2γ), and u2 is
linearly zero because the Jacobian matrix of the linear part of the system evaluated
in u2 is null. To decide the local behavior of u2 we must have a blow up. From now
on we fix l1 = γ, l2 = A+ 2γ.

After translating the singular point u2 to the origin, making the change of coor-
dinates u = U , v = UW and rescaling the common factor U , we obtain

U̇ = (1/2)U(AUW 2 + (A+ 2γ)U + 2UW + 4W + 2A+ 4γ), Ẇ = −W (W + γ).

Note that such system has two singularities when l1l2 6= 0, namely, U1 = (0, 0) and
U2 = (0,−γ); one singular point when l1 = 0 and l2 6= 0, namely U1 = U2. The
eigenvalues of U1 are −γ and A + 2γ, whereas the eigenvalues of U2 are A and γ.
From the combination of the signs of l1 and l2, as described in Figure 9, we obtain
the possible local behavior of U1 and U2.

Figure 9. The possible combination of signs of l1 and l2 describe
the cases to be considered for system (H.3) when c = 0.

After blowing down we obtain all possible phase portraits for system (H.3) when
c = 0. Note that each one is realizable. Indeed, the phase portrait HL.3.2 cor-
responds to subcase (1.1) which is realizable with A = 4 and γ = −1; HL.3.3
corresponds to subcase (1.2) which is realizable with A = 1 and γ = −1. Notice
that if γ 6= 0 there is a third invariant straight line, given by f3 = γ(x− y)− 1 = 0
so HL.3.3 is the only possible phase portrait for subcase (1.2). The phase portraits
HL.3.4 and HL.3.5 correspond, respectively, to subcases (2.1) and (3.1). The phase



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 25

portrait HL.3.4 is realizable with A = 1 and γ = 1, and HL.3.5 is realizable with
A = 1 and γ = 0.

Figure 10. On the left the local phase portrait after blow up.
Here they are indexed according to the signs of l1 and l2. On the
right the local behavior at origin after the Blow down for system
(H.3).

It remains to consider l2 = 0. With this condition the infinity is filled up of
singular points. After eliminating the common factor v we have only one singular
point at the local chart U1. The eigenvalues associated to this point are 2 and 1,
so this is an unstable node. By continuity the only possible phase portrait in this
case is HL.3.1 of Figure 2, which is realizable with A = 2 and γ = −1.

Now considering system (H.3) with β+γ = 0 we have seen above that the system
has a Darboux invariant when β = γ = 0 and α 6= 0. Under these conditions the



26 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

differential system is

ẋ = (A/2)(x2 − y2 − 1)− αy(1− cx− cy),
ẏ = (A/2)(x2 − y2 − 1) + cα(y2 + 1)− αx(1− cy).

(7.7)

Such system has f1 = x2− y2− 1 = 0 and f2 = y− x+ c = 0 as algebraic invariant
curves. If c = 0 then we obtain system (7.6) when γ = 0, so we can take c 6= 0 here.
Moreover, doing the transformation x = −X, y = −Y in the algebraic cubic we can
assume c > 0. Finally, since α is different from zero we can take α = 1 in (7.7).

System (7.7) has two finite singular points, namely z1 = ((2c − A)/2, −A/2)
and z2 = ((c2 + 1)/(2c), (1 − c2)/(2c)). Defining l1 = c2 − Ac − 1, l2 = A − c
and l3 = A − 2c, we have z1 coalesces with z2 if and only if l1 = 0. Moreover the
eigenvalues associated to z1 are l1 and 1, and the eigenvalues associated to z2 are
−l1 and 1. So we conclude that z1 is a unstable node and z2 is a saddle if l1 > 0;
z1 is a saddle and z2, an unstable node, if l1 < 0 and, if l1 = 0, z1 = z2 is a
saddle-node.

In the local chart U1, system (7.7) becomes

u̇ =
1

2
((A− 2c)u3 −Au2 +Auv2 − u(A− 2c)− v2(A− 2c) + 2u2v − 2v +A),

v̇ =
1

2
v((A− 2c)u2 +Av2 − 2cu+ 2uv −A),

which has three singularities u1 = (−1, 0) and u2 = (1, 0) and u3 = ( A
A−2c , 0), if

A 6= 2c. Note that when l3 = 0 the point u3 does not exist and u1 = u3 when l2 = 0.
The eigenvalues associated to u1 are 2l2 and 0, the point u2 has both eigenvalues
equal to −2c, and u3 has eigenvalues 0 and 2cl2/l3. It is not difficult to see that
when l2 6= 0, u1 and u3 are saddle–nodes. In the local chart U2 the origin (0, 0) is
a singular point if and only if l3 = 0.

Assuming l1l2 6= 0 and considering all possible combinations of the sign of l1, l2
and l3 we observe that there are some impossible combinations, for instance when
l2 < 0 we have l3 < 0. In Figure 11 we describe the possible combinations and
introduce a label for each one.

Figure 11. The possible combinations of signs of l1, l2 and l3 for
system (H.3) when c 6= 0.

The case (2.2.1) presents a unique phase portrait, HL.3.6 of Figure 2 and it is
realizable with A = 1/2 and c = 1.

In case (2.1.1) we have three possibilities for the finite saddle separatrix ω-limit
set: we can have a connection of separatrix as in HL.3.7; the separatrix can go to
the stable node, generating a phase portrait equivalent to HL.3.6, or the separatrix
can go to the parabolic sector of the saddle node u3 which corresponds to HL.3.8.
Moreover HL.3.8 is realizable with A = 2 and c = 1/2, and as we see above, HL.3.6



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 27

is realizable with A = 1/2 and c = 1. Since HL.3.6 and HL.3.8 are realizable then
by continuity of the parameters we conclude that HL.3.7 is also realizable.

The analysis of case (2.1.2) can be done as the case (2.1.1) and it has the phase
portraits equivalent to them.

The possible phase portraits of (2.1.3) are also equivalent to the phase portraits
of (2.1.1). Also the case (1.1.1) has a phase portrait equivalent to (2.2.1).

When l2 = 0 it follows that l1, l3 < 0 and in the local chart U1 the singular point
u1 = u3 is non-elementary. After translate this singular point to the origin, making
the change of coordinates u = U , v = UW and rescaling the common factor U we
obtain

U̇ = (U/2)(AUW 2 −AU + 2UW − 4W + 2A), Ẇ = W (W −A).

This system has two singularities U1 = (0, 0) and U2 = (0, A) being both saddles.
Figure 12 shows the blow down.

Figure 12. The local phase portrait of system (7.7) in the local
chart U1 when l2 = 0. On the left the local phase portrait after
blow up. On the right the local behavior at the origin after blow
down.

To obtain the phase portrait for system (7.7) with l2 = 0 we note that there is
more two invariant straight lines, given by f3 = x + y = 0 and f4 = Ax + Ay − 1.
The finite saddle z1 is on f3 = 0 and the finite node is on the intersection of f2 = 0
and f4 = 0 so by continuity there is only one phase portrait, which is topologically
equivalent to HL.3.3.

Finally it remains to study the case l1 = 0. Here l2 < 0 and l3 < 0 so the only
possibility is the phase portrait HL.3.9 of Figure 2, which is realizable with A = 0
and c = 1.

To conclude the proof of Proposition 7.2 it remains to study the global phase
portrait of system (H.4) when A = 2β and α 6= 0. In this case we assume α = 1,
so (H.4) is written as

ẋ = βx2 + (aβ −
√
a2 − 1)xy + (a−

√
a2 − 1β)y − β,

ẏ = (aβ −
√
a2 − 1)y2 + βxy + (a−

√
a2 − 1β)x+ (aβ −

√
a2 − 1).

Denoting δ = aβ −
√
a2 − 1 and η = a −

√
a2 − 1β there are at most three finite

singularities z1 = (−δ/η, β/η), z2 = ((δη − β)/(β2 − δ2), (δη − β)/(β2 − δ2)) and
z3 = ((β+δη)/(β2−δ2),−(β+δη)/(β2−δ2)). We observe that such singular points
never coalesce but if η = 0, z1 does not exist and if β2 − δ2 = 0 the same happens
with z2 or z3. With respect to the localization of these points, z3 is the intersection
of the hyperbola and the straight line, z1 is on the straight line and z2 is on the



28 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

hyperbola. Moreover it is not difficult to check that z1, z2 and z3 are hyperbolic
points, being z1 a saddle, z2 a stable node and z3 an unstable node.

Concerning to the behavior at infinity, in the local chart U1 the compactified
system is given by

u̇ = v(η − ηu2 + βuv + δv), v̇ = −v(β − βv2 + ηuv + δu),

so v is a common factor what means that v = 0 is a line of singular points. Elimi-
nating this common line it remains singularities if and only if η = 0 or β2− δ2 = 0.
When η = 0 the point u1 = (−a, 0) is a saddle. When δ = β the point u2 = (−1, 0)
is a node with eigenvalues η and 2η. Finally if δ = −β then the point u3 = (1, 0)
has eigenvalues −η and −2η so it is a node.

In the local chart U2, the system becomes

u̇ = −v(−η + βv + ηu2 + δuv), v̇ = v(δ + βu+ ηuv + δv2).

So eliminating the common factor v the origin is not a singular point.
By the previous study and continuity of the solutions we conclude that there

exist three possible phase portraits and they are topologically equivalent to the ones
obtained from system (H.2) and described in Figure 2. Indeed when η, β2 − δ2 6= 0
we have the phase portrait HL.2.1, when β2− δ2 = 0 we have HL.2.2, and the case
η = 0 corresponds to phase portrait HL.2.3.

Before to study the systems of type (P), we present two lemmas that will help
to show the realization or not of the phase portraits that follow.

Lemma 7.3. On any straight line which is not composed of orbits the total number
of contact points is at most two for any quadratic system. If there are two such
points p1 and p2, then the orbits intersecting the segment ∞p1 cross in the same
sense as the orbits intersecting p2∞, and the opposite sense to the path intersecting
p1p2.

Lemma 7.4. The straight line connecting one finite singular point and a pair of
infinite singular points in a quadratic system is either formed by trajectories or a
line with exactly one contact point. If this contact point is the finite singular point,
the flow goes in different directions on each half straight line.

The proof of Lemma 7.3 is in [8]. Lemma 7.4, in the case that the pair of infinite
singular points are saddles is in [16]. When such a pair are saddle-nodes, the proof
appeared in [1].

Proposition 7.5. Each real planar quadratic differential system with an invariant
parabola and an invariant straight line having a Darboux invariant can be written,
after an affine change of coordinates, as

(i) (P.1) with α1 − 2α2 6= 0 and Darboux invariant

I7(x, y, t) = e(α1−2α2)t(y − x2)−1x2.

(ii) (P.2) with α2(β1 − β2) 6= 0, γ2 = c = 0 and Darboux invariant

I8(x, y, t) = e2α2(β1−β2)t(y − x2)β2y−β1 ,

(iii) (P.2) with c γ2 6= 0, β1 = β2, α2 = 0 and Darboux invariant

I9(x, y, t) = e−2cγ2t(y − x2)(y + c)−1,

Moreover there are 41 non-equivalent phase portrait in the Poincaré disc
for these systems. They are in Figures 3 and 4.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 29

Proof. We fix the invariant parabola as f1 = y−x2 = 0. Here we describe in details
the proof of the existence of a Darboux invariant for system (P.2), the other cases
are analogous. System (P.2) is given by

ẋ = −β1(y−x2) + y(β2 + γ2x) + (α2 + γ2c)x+ cβ2, ẏ = 2(y+ c)(α2 + β2x+ γ2y),

where c α2 = 0. If c = 0 then the additional invariant line is written as f2 = y = 0
and if α2 = 0, such line is f2 = y + c = 0.

System (P.2) has a Darboux invariant if there exist λ1, λ2 not all zero satisfying
equation (3.3) with s ∈ R\{0}, and k1, k2 being the cofactors of f1 = 0 and f2 = 0,
respectively. For c = 0, k1 = 2(α2 + β1x + γ2y) and k2 = 2(α2 + β2x + γ2y).
Equation (3.3), with s 6= 0 has the solution

s = −2α2(λ1 + λ2), β2 = −β1λ1/λ2, γ2 = 0,

Taking λ1 = β2 and λ2 = −β1 the solution can be rewritten as

s = −2α2(β2 − β1), λ1 = β2, λ2 = −β1, γ2 = 0,

and the Darboux invariant is

I8(x, y, t) = e2α2(β1−β2)t(y − x2)β2y−β1 .

In this case we assume β2 − β1 6= 0 otherwise system (P.2) has a common factor.
Moreover if α2 = c = 0 (P.2) does not admit a Darboux invariant.

When α2 = 0 then f2 = y + c and the cofactors of f1 = 0 and f2 = 0 are,
respectively, k1 = 2(cγ2 + β1x+ γ2y) and k2 = 2(β2x+ γ2y). In this case equation
(3.3) has only one solution

s = −2cγ2λ1, β2 = β1, λ2 = −λ1.

So taking λ1 = 1 we obtain the Darboux invariant

I9(x, y, t) = e−2cγ2t(y − x2)(y + c)−1.

From now on we study the possible global phase portraits for systems (P) when
they have a Darboux invariant. We start studying system (P.1). Remember that
such system is given by

ẋ = x(α2 + β2x+ γ2y), ẏ = α1(y − x2) + 2α2x
2 + 2y(β2x+ γ2y).

We consider two cases: γ2 6= 0 and γ2 = 0. If γ2 6= 0 we assume γ2 = 1. In this last
case system (P.1) have at most four singular points, given by

z1 = (0, 0), z2 = (0,−α1/2),

z3 =
(
− (β2 +

√
β2

2 − 4α2)/2, (β2
2 − 2α2 + β2

√
β2

2 − 4α2)/2
)
,

z4 =
(
− (β2 −

√
β2

2 − 4α2)/2, (β2
2 − 2α2 − β2

√
β2

2 − 4α2)/2
)
.

Observe that applying the change of coordinates x = −X, y = Y we can assume
β2 ≥ 0. Let l1 = α1, l2 = α2, l3 = β2

2 − 4α2 − β2

√
β2

2 − 4α2 and l4 = α1 − 2α2 be.
It follows from Proposition 7.5 (i) l4 6= 0. Moreover

• z1 has eigenvalues l1 and l2;
• z2 has eigenvalues −l1 and −l4;
• z3 has eigenvalues l4 and (β2

2 − 4α2 + β2

√
β2

2 − 4α2)/2;
• z4 has eigenvalues l3 and l4,



30 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

so l21 + l22 6= 0 and the topological type of the finite singular points can be studied
using the Hartman-Grobman Theorem and [9, Theorem 2.19].

With respect to the position of the finite singularities, z1 is on the intersection
of the parabola and the straight line, z2 is on the straight line, and z3, z4 are on
the parabola.

In the local chart U1, system (P.1) is written as

u̇ = u2 + β2u+ (α1 − α2)uv + 2α2 − α1, v̇ = −v(α2v + u+ β2),

which has at most two singular points when v = 0, namely

u1 = (−β2 −
√
β2

2 + 4(α1 − 2α2)/2, 0), u2 = (−β2 +
√
β2

2 + 4(α1 − 2α2)/2, 0).

The eigenvalues associated to u1 are −
√
β2

2 + 4l4 and −(β2 −
√
β2

2 + 4l4)/2 while

the eigenvalues associated to u2 are
√
β2

2 + 4l4 and −(β2 +
√
β2

2 + 4l4)/2.
Since we are assuming β2 ≥ 0 it follows that when β2

2 + 4l4 > 0 the point u2 is a
saddle and it is not difficult to see that if l4 > 0, then u1 is a saddle, and if l4 < 0,
u1 is a stable node. When β2

2 + 4l4 = 0 u1 and u2 coalesce and we conclude that
this point is a saddle-node, using [9, Theorem 2.19]. When β2

2 + 4l4 < 0 there is no
infinite points in the local chart U1.

In the local chart U2 the origin (0, 0) is a stable node.
Observe that l1, l2, l3, l4, β

2
2−4α2 and β2

2 +4l4 are bifurcation surfaces, i.e. where
topological changes in the global phase portrait of (P.1) can happen. To draw all
non-equivalent phase portraits of system (P.1) we split the study in three cases:
β2

2 − 4α2 > 0, β2
2 − 4α2 = 0 and β2

2 − 4α2 < 0.
Choosing a representative of each region defined by such surfaces we have a con-

figuration of finite and infinite points. Considering the behavior of the separatrices
of these systems we obtain all possible phase portraits when β2

2 − 4α2 > 0, thus
we obtain the 40 phase portraits described in Figures 13 and 14 and the phase
portraits 41− 50 of Figure 18. We study all these cases bellow.

Among the phase portraits 1−18 of Figure 13, we claim that 1 and 3, as well as 7
to 18, are not realizable. Indeed these 18 phase portraits, 1−3 present the possible
combinations when the singular points in the local chart U1 are both saddles. In
the finite part we have z1 and z3 unstable nodes, z2 is a stable node and z4 is a
saddle. So we have l1, l2, l4 > 0 and l3 < 0. In phase portrait 1 of Figure 13,
consider the straight line joining the finite singular point z3 to the infinity singular
point u1 as shows Figure 15. We can see that near the singular point z3 but on
opposite sides, the vector field has the same direction, which contradicts Lemma
7.4. So the phase portrait 1 of Figure 13 is not realizable. With the same argument
the portrait 3 of Figure 13 is also not realizable. So phase portrait 2 of Figure 13
is the only realizable and corresponds to phase portrait PL.1.1 of Figure 3.

Considering the phase portraits 4–18 of Figure 13 we shall prove that 7− 18 are
not realizable. First consider the phase portrait 7 and the straight line joining the
middle point between the infinity singular points u1 and u2 and the middle point
between the finite singular points z3 and z4 as shows Figure 16. By Lemma 7.3
this line should have at most two points of contact with the vector field, which does
not occur. In Figure 16 we can see at least four contact points, represented by the
smaller points that are not singularities of the system. This fact guarantees that
the ω−limit set of u2 is the finite point z4 on the parabola. So phase portraits 7−18
are not realizable using similar arguments. So among the phase portraits 4 − 18



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 31

Figure 13. Phase portraits of system (P.1) when γ2 = 1 and
β2

2 − 4α2 > 0.

only 4, 5 and 6 are realizable, which correspond, respectively to phase portraits
PL.1.2, PL.1.3 and PL.1.4 of Figure 3. The values of the parameters that realize
these systems can be found in Table 2.

The phase portraits 19−20 in Figure 13 and 21-26 in Figure 14 are topologically
equivalent to one of the phase portraits 1–18 in Figure 13 so they can be realizable



32 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Figure 14. Phase portraits of system (P.1) when γ2 = 1 and
β2

2 − 4α2 > 0.

or not, depends on their configuration. In Table 1 we present the relation among
the equivalent phase portraits of system (P.1) when c 6= 0. In the case where they
are topologically equivalent to a realizable phase portrait, we need not consider the
study again. However if they are topologically equivalent to a phase portrait which
was not realizable, we need to study it.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 33

Figure 15. The straight line joining the finite singular point z3

to the infinity singular point u1 in phase portrait 1 of Figure 13.

Figure 16. The straight line joining the middle point between the
infinity singular points u1 and u2 and the middle point between
the finite singular points z3 and z4 in phase portrait 7 of Figure
13.

Considering the same straight line used to prove the non-realization of phase
portraits 7–18 of Figure 13 we apply Lemma 7.3 to conclude that 21, 22, 25 and 26
of Figure 14 are not realizable.

The phase portraits 27–31 in Figure 14 present all the possibilities when there
are four finite singular points and one infinite singular point on the local chart
U1. Phase portraits 27–29 are realizable and correspond to phase portraits PL.1.5,
PL.1.6 and PL.1.7 of Figure 3. The values of the parameters that realize these
systems can be found in Table 2. Moreover 30 and 31 are topologically equivalent
to one of these three phase portraits.

Finally if there are four finite singular points and the local chart U1 has no
singular point we obtain the phase portraits 32 − 36 in Figure 14. For phase
portraits 32 and 33 of Figure 14 we consider the straight line x = z1

4 where the
finite singularity z4 is z4 = (z1

4 , z
2
4), and apply Lemma 7.4 to see that they are not

realizable (see Figure 17).

Figure 17. The straight line x = z1
4 = −(β2 −

√
β2

2 − 4α2)/2 in
phase portrait 32 of Figure 14.



34 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Table 1. Table of relations among all the possible phase portraits
of system (P.1) when c 6= 0.

Phase portrait Topologicaly equiv.
19 2
20 6
21 12
22 9
23 2
24 6
25 12
26 9
30 29
31 29
35 34
36 34
60 50

Moreover the phase portraits 35 and 36 are topologically equivalent to the phase
portrait 34 which is the only realizable phase portrait for this case and it is repre-
sented by PL.1.8 in Figure 3. The values of the parameters that realize this system
can be found in Table 2.

For β2
2 − 4α2 > 0 we consider the cases with three finite singular points. When

z1 = z2 the origin is a saddle-node and there are ten possible phase portraits,
namely 37–40 in Figure 14 and 41–46 in Figure 18. But since the nodal sector of
the saddle node must have its orbits tangent to its separatrix, the phase portraits 37
and 38 in Figure 14 are not realizable. In other words the separatrices of the saddle-
node z1 must be on the invariant parabola. With the same argument the phase
portraits 41, 42, 45 and 46 of Figure 18 also are not realizable. So when z1 = z2

the realizable phase portraits are 39, 40, 43 and 44 of Figure 18, corresponding to
PL.1.9, PL.1.10, PL.1.11 and PL.1.12 in Figure 3, respectively. The values of the
parameters that realize these systems can be found in Table 2.

When there are three finite singularities with z1 = z4 then by continuity we have
the phase portraits 47–50 of Figure 18. All these for phase portraits are realizable
and correspond, to PL.1.13, PL.1.14, PL.1.15 and PL.1.16 in Figure 3, respectively.
The values of the parameters that realize these systems can be found in Table 2

For β2
2 − 4α2 = 0 there is another case with three finite singularities that corre-

spond to the case z3 = z4. Here we can have ten phase portraits, given by 51− 60
in Figure 18. The phase portraits 51, 52 and 55 are realizable and corresponds,
respectively, to PL.1.17, PL.1.18 and PL.1.19 in Figure 3. The values of the pa-
rameters that realize these systems can be found in Table 2. The phase portraits 53
and 54 are not realizable. The idea again is to use Lemma 7.4 with the straight line
joining the origin of the local chart U3 to the singular point u2 of the local chart
U1. By Figure 19 and this lemma the phase portraits 53 and 54 are not realizable.

Considering the phase portraits 56 and 57 we will show that they are not realiz-
able. Take the straight line passing through the origin of the local chart U1 and the
infinite singular point u1 = u2 (see Figure 20). The contact points on this straight
line contradicts Lemma 7.4 so the phase portraits 56 and 57 are not realizable.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 35

Figure 18. Phase portraits 41−50 corresponds to phase portraits
of system (P.1) when γ2 = 1 and β2

2 − 4α2 > 0; Phase portraits
51−60 corresponds to phase portraits of system (P.1) when γ2 = 1
and β2

2 − 4α2 = 0.



36 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Figure 19. The straight line connecting the origin of the local
chart U3 with the singular point u2 of the local chart U1 in phase
portrait 53 of Figure 18.

About the phase portraits 58 and 59, considering the straight line passing through
the points z1 and z3 we have Figure 21 that is a contradiction with Lemma 7.3. So
they are not realizable. The phase portrait 60 is topologically equivalent to 50 of
Figure 18.

Figure 20. The straight line connecting the origin of the local
chart U3 with the singular point u1 = u2 of the local chart U1 in
phase portrait 56 of Figure 18.

Figure 21. The straight line passing through the points z1 and
z3 in phase portrait 58 of Figure 18.

If z3 = z4 and z1 = z2 we have the phase portraits 61, 62 and 63 of Figure 22.
But using the straight line joining z1 and z3 as done in Figure 21 and applying
Lemma 7.3 we see that 61 and 62 are not realizable. The phase portrait 63 is
realizable and corresponds to PL.1.20 in Figure 3. The values of the parameters
that realize this system can be found in Table 2.

For β2
2 −4α2 < 0 the points z3 and z4 are complex. The possible phase portraits

are described by 64 − 72 of Figure 22. The phase portraits 64, 65, 68 and 71 are
realizable and corresponds, respectively, to PL.1.21, PL.1.22, PL.1.23 and PL.1.24
of Figure 3. The values of the parameters that realize these systems can be found
in Table 2. To prove that the phase portraits 66, 67, 69 and 70 are not realizable, it



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 37

Figure 22. Phase portraits 61−63 corresponds to phase portraits
of system (P.1) when γ2 = 1 and β2

2 − 4α2 = 0; Phase portraits
64−72 corresponds to phase portraits of system (P.1) when γ2 = 1
and β2

2 − 4α2 < 0.

is enough to consider the straight line passing through the origin of the local chart
U3 and the infinity singularity u1 = u2 of the local chart U1 (see Figure 23). This
straight line generates a contradition with Lemma 7.4 so the phase portraits 66,
67, 69 and 70 are not realizable.

Figure 23. The straight line connecting the origin of the local
chart U3 with the singular poin u1 = u2 in the local chart U1 in
phase portrait 66 of Figure 22.

To end the case γ2 = 1 we consider the case where there is only one finite
singular point. Using [9, Theorem 2.19] we can see that the point is a saddle,
which generates phase portrait 72 of Figure 22 which corresponds to phase portrait



38 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

PL.1.25 of Figure 4. The values of the parameters that realize this system can be
found in Table 2.

Now we consider the case γ2 = 0. The system is

ẋ = x(α2 + β2x), ẏ = α1(y − x2) + 2x(α2x+ β2y).

When α1 = 0 such system has a common factor so assume α1 = 1. By the change
x = −X, y = Y it is enough to consider the case β2 ≥ 0. Assuming β2 > 0.
In the finite part the points z1 = (0, 0) and z2 =

(
−α2/β2, (α2/β2)2

)
are the

singular points and the system has an additional invariant straight line, given by
f3 = x+ α2/β2 = 0. Defining l1 = α2 and l2 = 1− 2α2 the eigenvalues associated
to z1 are 1 and l1, while the eigenvalues associated to z2 are −l1 and l2. We assume
l2 6= 0 (otherwise such system has a common factor and it is equivalent to a linear
system).

In the local chart U1 the unique singular point is u1 = (l2/β2, 0) and it is a
saddle. In the local chart U2 the compactified system is

u̇ = u((1− 2α2)u2 + (α2 − 1)v − β2u), v̇ = v((1− 2α2)u2 − 2β2u− v).

The origin (0, 0) is a linearly zero singularity. Doing the blow up u = UV, v = V
and rescaling by V we obtain the system

U̇ = U(α2 + β2U), V̇ = V ((1− 2α2U
2V )− 2β2U − 1).

When V = 0 the singularities are u1 = (0, 0) and u2 = (−α2/β2, 0). The eigenvalues
associated to u1 are −1 and l1 while the eigenvalues of u2 are −l1 and −l2. The
blowing down process is described in Figure 24 (1)-(4) according to the signs of l1
and l2.

When β2 = 0 the point z1 is the unique finite singular point, being a saddle or
an unstable node depending on the sign of l1. In the local chart U1 there is no
singular point and the origin (0, 0) of U2 is linearly zero. To study such point we
apply blow ups, in Figure 24 is described the blowing down (5) and (6).

Summarizing the study done previously we obtain the local behaviour at origin
of U2:

(1) β2 > 0, l1 > 0 and l2 > 0: the origin of U2 has two elliptic sectors;
(2) β2 > 0, l1 > 0 and l2 < 0: the origin of U2 has two hyperbolic sectors;
(3) β2 > 0, l1 < 0 and l2 > 0: the origin of U2 has two elliptic sectors;
(4) β2 > 0, l1 = 0 and l2 > 0: the origin of U2 has two elliptic sectors.
(5) β2 = 0, l1 > 0: the origin of U2 has two hyperbolic sectors;
(6) β2 = 0, l1 < 0: the origin of U2 has two elliptic sectors.

By continuity and the above analysis we conclude that the case (3) is topolog-
ically equivalent to case (1) and the cases (1), (2), (4), (5) and (6) correspond,
respectively, to the phase portraits PL.1.26, PL.1.27, PL.1.28, PL.1.29 and PL.1.30
of Figure 4. Table 4 has the values of the parameters that realizes the phase por-
traits of system (P.1)

System (P.2) with c 6= 0 has a Darboux invariant if γ2 6= 0, and it can be written
as

ẋ = β1(x2 + c) + γ2x(y + c), ẏ = 2(y + c)(β1x+ γ2y).

Note that if β1 = 0 such system has a common factor so we can assume β1 = 1.
Applying the change of coordinates x = −X, y = Y and rescaling the time we can
assume γ2 > 0.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 39

Figure 24. Blow down of system (P.1) when γ2 = 0.



40 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Table 2. Table of values for the parameters of system (P.1).

γ2 β2 α2 α1

PL.1.1 1 1 1/8 1
PL.1.2 1 1 1/16 1/16
PL.1.3 by continuity
PL.1.4 1 1 1/16 1/150
PL.1.5 1 1/2 3/64 1/32
PL.1.6 by continuity
PL.1.7 1 1 -3/8 -1
PL.1.8 1 1 3/16 1/16
PL.1.9 1 1 1/16 0
PL.1.10 1 1 -1 0
PL.1.11 1 1 1/18 0
PL.1.12 1 1 3/16 0
PL.1.13 1 1 0 1
PL.1.14 1 1 0 -1/8
PL.1.15 1 1 0 -1/4
PL.1.16 1 1 0 -1
PL.1.17 1 1 1/4 1
PL.1.18 1 1 1/4 3/8
PL.1.19 1 1 1/4 1/4
PL.1.20 1 1 1/4 0
PL.1.21 1 1 2 5
PL.1.22 1 3 4 6
PL.1.23 1 1 9/8 2
PL.1.24 1 1 2 13/4
PL.1.25 1 1 2 0
PL.1.26 0 1 1/4 1
PL.1.27 0 1 3/2 1
PL.1.28 0 1 0 1
PL.1.29 0 0 1 1
PL.1.30 0 0 -1 1

Figure 25. Local phase portraits



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 41

If c < 0 the system has three finite singular points z1 = (−1/γ2, 1/γ
2
2), z2 =

(−
√
−c,−c) and z3 = (

√
−c,−c). Otherwise, only z1.

Defining l1 = c 6= 0 and l2 = 1 + c γ2
2 the eigenvalues associated to z1 are 2γ2l1

and l2/γ2, the eigenvalues associated to z2 are −2
√
−c and −2(γ2 c +

√
−c); the

eigenvalues associated to z3 are 2
√
−c and −2(γ2 c −

√
−c). So when c < 0 the

point z3 exists and it is an unstable node.
In the local chart U1 we have two singular points u1 = (0, 0) being a hyperbolic

saddle and u2 = (−1/γ2, 0) being a saddle-node. In the local chart U2 the origin is
a stable node.

When l2 = 0 then z1 = z3 is a semi-hyperbolic node and the infinity part does
not change. Note that z1 is a saddle-node in this case. So by continuity and
the reasoning above, if c > 0 we have phase portrait PL.2.1 of Figure 4 which is
realizable with c = γ2 = 1. When c < 0 and l2 6= 0 the system has two possible
phase portraits, also described in Figure 4: PL.2.2 (realizable with c = −1/2 and
γ2 = 1) and PL.2.3 (realizable with c = −2 and γ2 = 1).

Finally if c < 0 and l2 = 0, we see that the line y + c = 0 is one of the separatix
of the saddle-node. So the only possible phase picture is PL.2.4 (realizable with
c = −1 and γ2 = 1).

Now we study the global phase portraits of systems (P.2) when c = 0 and they
have a Darboux invariant. The differential system is

ẋ = −β1(y − x2) + β2y + α2x, ẏ = 2y(β2x+ α2).

Since α2 6= 0 we take α2 = 1. Moreover doing the change of coordinates x =
−X, y = Y we can assume β2 ≥ 0. The system has at most three finite singular
points, namely, z1 = (0, 0) and z2 = (−1/β1, 0) and z3 = (−1/β2, 1/β

2
2). The

point z1 has eigenvalues 2 and 1, so it is an unstable node. On the other hand the
topological type of z2 and z3 depends on the numbers l1

.
= β1 and l2

.
= β1−β2 6= 0.

Indeed the point z2 has eigenvalues −1 and 2l2/l1 and z3 has the eigenvalues −1
and −2l2/l1.

In the local chart U1 the system has u1 = (0, 0) as a singularity with eigenvalues
−l1 and −l3, where l3 = β1 − 2β2.

In the local chart U2 the compactified system has the origin as a nilpotent singu-
larity. This mean that the linear part of the system, evaluated in (0, 0), is not null
but their eigenvalues are both equal to zero. To classify this type of singular point
we use Theorem 3.5 of [9]. This result use two functions, F (u) = aMu

M + o(uM )
and G(u) = bNu

N + o(uN ), defined from the differential system. In short the
caracterization is done using aM , bN and the natural numbers M,N .

For the compactified system in the local chart U2 these functions are

G(u) = −2(β2 − 3β2)

l2
u+

5l3
l22
u2, F (u) =

2β2l3
l22

u3 +
2l23
l32

u4.

So when l3 > 0 the origin (0, 0) is a saddle as in (b) of Figure 25. If l3 < 0 the
origins consists of one hyperbolic and one elliptic sector as in (a) of Figure 25.

By continuity, when l1 > 0 and l3 > 0 we have the phase portrait PL.2.5 of
Figure 4 (realizable with β1 = 4 and β2 = 1). If l3 < 0 we have the phase portraits
PL.2.6 (realizable with β1 = 3/2 and β2 = 1) and PL.2.7 (realizable with β1 = 1/2
and β2 = 1) of Figure 4. Now if l1 < 0 the only possibility is l3 < 0 and we have
the phase portrait PL.2.8 (realizable with β1 = −1 and β2 = 1) of Figure 4.



42 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

If l1 = 0 the point z2 goes to the infinity and collide with u1 becoming a saddle-
node. Moreover l1 = 0 implies l3 < 0, so the origin of U2 has a hyperbolic and
one elliptic sector. This case corresponds to phase portrait PL.2.9 of Figure 4,
realizable with β1 = 0 and β2 = 1.

If β2 = 0 the point z3 goes to the infinity and collide with the origin of U2

becoming (0, 0) a nilpotent saddle-node as (c) or (d) in Figure 25. Moreover the
only possible phase portrait is given by PL.2.10 of Figure 4, realizable with β1 = 1
and β2 = 0).

Finally when l3 = 0 then the infinity if filled of singular points, without special
singularities and the corresponding phase portrait is PL.2.11 of Figure 4 (realizable
with β1 = 2 and β2 = 1). �

Proposition 7.6. Each real planar polynomial differential system with two invari-
ant real lines that intersect at a single point and a third invariant straight line
having a Darboux invariant can be written, after an affine change of coordinates,
as

(i) (LV.1) with α(q − β) 6= and Darboux invariant

I10(x, y, t) = eα(q−β) tyβxβ−q+r(y − x)−(β+r),

(ii) (LV.2) with c = q = 0, p 6= 0 and Darboux invariant

I11(x, y, t) = e−ptxy−r,

(iii) (LV.2) with c = −1 and Darboux invariant

I12(x, y, t) = ety (y − 1)−1,

(iv) (LV.3) with α = −(β + 1), c β 6= 0 and Darboux invariant

I13(x, y, t) = e−c β ty (y + ax+ c)−1.

Moreover there are 27 non-equivalent phase portraits in the Poincaré disc. They
are in Figure 5.

Proof. Let f1 = x = 0, f2 = y = 0 be the two real straight lines intersecting in
a point. Considering system (LV.1) the third line is f3 = y − x and the cofactors
associated to f1, f2 and f3 are, respectivelly, k1 = α+ ry+ βx, k2 = α+ y(β − q +
r)+qx and k3 = α+y(β−q+r)+βx. One solution for equation λ1k1 +λ2k2 +s = 0
is

λ2 =
βλ1

β − q + r
, λ3 = − (β + r)λ1

β − q + r
, s =

α(q − β)λ1

β − q + r
,

Taking λ1 = β − q + r we obtain the Darboux invariant

I10(x, y, t) = eα(q−β) tyβxβ−q+r(y − x)−(β+r).

Now we analyze system (LV.2) that has f3 = y + c as the third invariant straight
line(remember that c = 0 or c = −1). Here the cofactors are k1 = p + qx + ry,
k2 = y + c and k3 = y. If c = 0 then equation (3.3) has only one the solution

q = 0, λ3 = −rλ1 − λ2, s = −pλ1.

Taking λ1 = 1 we obtain the Darboux invariant

I11(x, y, t) = e−ptxy−r.

Otherwise if c = −1 then the more general solution is

λ1 = 0, λ3 = −λ2, s = λ2.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 43

Taking λ2 = 1 we obtain the Darboux invariant

I12(x, y, t) = ety(y − 1)−1.

The last case to be considered is system (LV.3) that has f3 = y+ ax+ c = 0 as the
third straight line. The cofactors are k1 = −α(y+ax+c)−y, k2 = β(y+ax+c)+ax
and k3 = βy − aαx. Solving equation (3.3) we obtain the solution

α = −(β + 1), λ2 = −λ1 − λ2, s = −c(λ1 + β(λ1 + λ2)).

Taking λ1 = 0 and λ2 = 1 then we obtain the Darboux invariant

I13(x, y, t) = e−c β ty (y + ax+ c)−1.

We begin the study of the global phase portraits with systems (LV.1) when they
have a Darboux invariant. Remember that if system (LV.1) has a Darboux invariant
then β − q 6= 0 and α 6= 0 so we can take α = 1 getting

ẋ = x(1 + βx+ ry), ẏ = y(1 + qx+ (β − q + r)y). (7.8)

Define l1 = (β − q)/(β − q+ r), l2 = (β − q)/β and l3 = (β − q)/(β + r). The finite
part presents at most four singularities

• z1 = (0, 0) with eigenvalues both equal to 1;
• z2 = (0,−1/(β − q + r)) with eigenvalues −1 and l1;
• z3 = (−1/β, 0) with eigenvalues −1 and l2;
• z4 = (−1/(β + r), −1/(β + r)) with eigenvalues −1 and −l3.

In the local chart U1 the compactified system has two singular points, being
u1 = (0, 0) with eigenvalues −β and −(β−q) and u2 = (1, 0) with eigenvalues β−q
and −(β + r). Moreover in the local chart U2 the origin (0, 0) is a singular point
with eigenvalues −(β−q) and −(β−q+r). Thus when one of the finite singularities
goes to infinity, it collides with u1, u2, or the origin of the local chart U2.

When l1, l2 and l3 are non-zero, the combinations between their signs generate
the possible phase portraits of system (7.8). There are exactly three possible phase
portraits, all of them described in Figure 5: LVL.1.1, realizable for β = 1, q =
r = 0; LVL.1.2, realizable for β = 1, q = r = −2; LVL.1.3, realizable for β = 1,
q = −r = 3/4.

Now we consider the case β = −r 6= 0. Here only the point z4 goes to the
infinity and collides with u2 making it a semi hyperbolic saddle-node. There are
two possible phase portraits, given by LVL.1.4 of Figure 5 (realizable with β = 1,
q = r = −1) and LVL.1.5 of Figure 5 (realizable with β = 2, q = 1, r = −2). The
cases where z2 or z3 goes to the infinity generate phase portraits equivalent to the
previous ones.

Finally when two finite singular points go to the infinity (for example when
β = −r and q = 0), then there is only one phase portrait, given by LVL.1.6 of
Figure 5. This last phase portrait is realizable for β = 1, q = 0 and r = −1.

Now we consider the systems (LV.2) when they have a Darboux invariant we
split in two cases. First we consider the case c = −1, when the system is given by

ẋ = x(p+ qx+ ry), ẏ = y(y − 1).

If q 6= 0 unless of the change x = X/q we can assume q = 1. Considering q = 1
and defining l1 = p, l2 = −(p+ r) and l3 = r− 1 the system has at most four finite
singular points, namely

• z1 = (0, 0) with eigenvalues −1 and l1;



44 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

• z2 = (0, 1) with eigenvalues 1 and −l2;
• z3 = (−p, 0) with eigenvalues −1 and −l1;
• z4 = (−p− r, 1) with eigenvalues 1 and l2.

In the local chart U2 the origin (0, 0) is a singularity with eigenvalues −1 and
l3. In the local chart U1 the system has two singularities if l3 6= 0: u1 = (0, 0)
being a hyperbolic unstable node and u2 = (1/l3, 0) with eigenvalues 1 and 1/l3.
Hence if l3 = 0 the point u2 collides with the origin of U2 making it a semi-
hyperbolic singularity of type saddle node. By continuity and using all the possible
combinations of the signs of l1, l2 and l3 when q = 1 and l3 6= 0 we obtain the phase
portraits LVL.2.1–LVL.2.7 of Figure 5. When l3 = 0, i.e., r = 1 has three possible
phase portraits: LVL.2.8, LVL.2.9 and LVL.2.10 of Figure 5. The values of the
parameters that realize these systems can be found in Table 3. Now it remains to
study the case q = 0. Note that since the system cannot have common factors it
follows that l1 and l2 are different from zero. When q = 0 both the finite part and
the analyzes in the local chart U2 remain almost the same. The only difference in
the finite part is that the singularities z3 and z4 go to infinity. However in the local
chart U1 the compactified system is

u̇ = −u((r − 1)u+ (p+ 1)v), v̇ = −v(pv + ru).

So the origin is a linearly zero singular point if l3 6= 0 and we apply the blow up
doing the change of coordinates u = U, v = UW . The new system is

U̇ = −U2((p+ 1)W + r − 1), Ẇ = UW (W − 1).

After eliminating the common factor U it remains two singular points on U = 0:
u1 = (0, 0) with eigenvalues −1 and −l3, and u2 = (0, 1) with eigenvalues 1 and l2.
Hence they are hyperbolic points and doing the blow down the origin of U2 has (for
l3 6= 0)

• two elliptic sectors if u1 is a saddle and u2 is a unstable node. This case
corresponds to phase portrait LVL.2.11 of Figure 5;
• two elliptic sectors if u1 is a stable node and u2 is a saddle. This case

corresponds to phase portrait LVL.2.12 of Figure 5;
• two parabolic sectors if u1 and u2 are both saddles and there is a saddle and

a node as singular finite points. This case corresponds to phase portrait
LVL.2.13 of Figure 5;
• two parabolic sectors if u1 and u2 are both saddles and there are two nodes

as singular finite points. This case corresponds to phase portrait LVL.2.14
of Figure 5;
• six parabolic sectors if u1 and u2 are both saddles and there are two nodes

as singular finite points. This case corresponds to phase portrait LVL.2.15
of Figure 5.

The last possibility when c = −1 is q = 0 and l3 = 0. But when this happens the
system has the infinity line v = 0 filled up of singular points. After eliminating the
common factor v, in the local chart U1 the point u1 = (0, 0) is a singular point,
with eigenvalues −l1 and l2. In the local chart U2, After eliminating the common
factor v, the origin is a singularity. By continuity the possible phase portraits are
LVL.2.16 and LVL.2.17 of Figure 5. In Table 3 we put the values of the parameters
that realizes each one of the phase portraits described in Figure 5.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 45

Table 3. Table of values for the parameters of system (LV.2) when
c = −1.

q r p
LVL.2.1 1 -1 1/2
LVL.2.2 1 2 1
LVL.2.3 1 -1 2
LVL.2.4 1 -1 1
LVL.2.5 1 2 -2
LVL.2.6 1 1/2 -1/2
LVL.2.7 1 0 0
LVL.2.8 1 1 1
LVL.2.9 1 1 -1/2
LVL.2.10 1 1 -1
LVL.2.11 0 -2 1
LVL.2.12 0 2 1
LVL.2.13 0 0 1
LVL.2.14 0 2 -1
LVL.2.15 0 3/4 -1/4
LVL.2.16 0 1 1
LVL.2.17 0 1 -1/2

Finally when c = 0 we obtain the differential system

ẋ = x2, ẏ = y(p+ rx), (7.9)

with p 6= 0. So we can take p = 1 and the system becomes a particular case of
system (DL) of Theorem 2.3. The global phase portraits of this system will be done
in the proof of Proposition 7.8 and the correponding phase portraits of system (7.9)
are described by DL.1, DL.2 and DL.3 of Figure 6.

To complete the proof of Proposition 7.6 we study the global phase portraits of
systems (LV.3). When (LV.3) has a Darboux invariant the parameter α must be
equal to −(β + 1) so the differential system is

ẋ = x(ax+ β(y + ax+ c) + c), ẏ = y(ax+ β(y + ax+ c)).

In the finite part there are three singular points, namely z1 = (0, 0), z2 = (0,−c)
and z3 = (−c/a, 0) (remember that a c 6= 0). Defining l1 = c β 6= 0 and l2 =
c(β + 1) 6= 0, then the eigenvalues of the z1 are l1 and l2; the eigenvalues of z2 are
c and −l1, and the eigenvalues associated to z3 are −c and −l2.

In the local chart U1 the compactified system becomes

u̇ = −cuv, v̇ = −v(cv + β(u+ cv + a) + a).

Hence the line v = 0 is filled of singular points after eliminating the common
factor v there are no singular points. The same happens in the local chart U2.
So by continuity the only possible phase portrait is LVL.3.1 of Figure 5, which is
realizable for β = 1 and a = c = −1. �

Proposition 7.7. Each real planar quadratic differential system with two parallel
real invariant straight lines and a third invariant straight line having a Darboux



46 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

invariant can be written, after an affine change of coordinates, as system (RPL)
and it has the Darboux invariant

I14(x, y, t) = e2t(x+ 1)(x− 1)−1.

Moreover there are 17 non-equivalent phase portraits in the Poincaré disc for this
system. They are described by RPL.1–RPL.17 in Figure 6.

Proof. Let f1 = x + 1 = 0, f2 = x − 1 = 0 and f3 = y = 0 be the three invariant
straight lines. The cofactors of f1, f2 and f3 are, respectivelly, k1 = x−1, k2 = x+1,
k3 = α + βx + γy. With these cofactors equation (3.3) with s ∈ R \ {0} has two
solutions, namely

s1 = {γ = 0, s = 2λ1 + (β − α)λ3, λ2 = −(λ1 + βλ3)}
s2 = {s = 2λ1, λ2 = −λ1, λ3 = 0}.

Since the second solution s2 is more general, we conclude that every quadratic
system that has two real parallel straight lines and a third real straight line as
invariant straight lines also has a Darboux invariant. Taking λ1 = 1 we obtain the
invariant

I14(x, y, t) = e2t(x+ 1)(x− 1)−1.

To draw the possible global phase portraits, remember that the system is

ẋ = x2 − 1, ẏ = y(α+ βx+ γy).

When γ 6= 0 we can take γ = 1 (indeed, just do the change x = X, y = Y/γ).
So the system can present at most four finite singularities, namely, z1 = (−1, 0),
z2 = (−1, β−α), z3 = (1, 0) and z4 = (1,−β−α). Define l1 = α−β and l2 = α+β.
The eigenvalues associated to z1 are −2 and l1 while the eigenvalues associated to z2

are −2 and −l1. Moreover z1 = z2 when l1 = 0. Analogously the eigenvalues of z3

are 2 and l2, while the eigenvalues associated to z4 are 2 and −l2, with z3 = z4 when
l2 = 0. So in the finite part the system can have two, three or four singularities,
depending on the values of l1 and l2.

In the local chart U1 the compactified system has at most two singularities on
the infinity line: u1 = (0, 0) and u2 = (1 − β, 0). Defining l3 = β − 1 we see that
u1 = u2 when l3 = 0 and the topological type of these singularities depends on
the sign of l3. Indeed the eigenvalues associated to u1 are −1 and l3 while the
associated to u2 are −1 and −l3.

In the local chart U2 we just need to check if the origin (0, 0) is a singularity,
which is true. It is a node, with the two eigenvalues equal to −1.

So considering γ 6= 0 and combining all the possibilities of the signs of l1, l2 and
l3 we obtain the phase portraits RPL.1–RPL.10 of Figure 6. In Table 3 we put the
values of the parameters that realizes each one of the phase portraits described in
Figure 6.

If γ = 0 then z2 and z4 goes to the infinity and the compactified system in the
local chart U2 becomes

u̇ = (1− β)u2 − αuv − v2, v̇ = −v(βu+ αv).

Note that when l3 = 0(β = 1) the line v = 0 is filled up of singular points, and
when l3 6= 0 the origin (0, 0) is a linearly zero singularity. Considering this case



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 47

Table 4. Table of values for the parameters of system (RPL).

α β γ
RPL.1 -5/4 1/4 1
RPL.2 0 -1 1
RPL.3 -3 2 1
RPL.4 -2 1 1
RPL.5 0 1 1
RPL.6 -1/2 1/2 1
RPL.7 1/2 -1/2 1
RPL.8 -2 2 1
RPL.9 -1 1 1
RPL.10 0 0 1
RPL.11 -3 2 0
RPL.12 0 -1 0
RPL.13 -1 0 0
RPL.14 -1 2 0
RPL.15 -1/4 3/4 0
RPL.16 -2 1 0
RPL.17 0 1 0

first and applying the blow up u = U, v = UW and dividing by U we obtain the
system

U̇ = −U(β +W 2 + αW − 1), Ẇ = W (W − 1)(W + 1). (7.10)

When U = 0 the singularities of (7.10) are u1 = (0,−1) with eigenvalues 2 and l1,
u2 = (0, 0) with eigenvalues −1 and −l3, and u3 = (0, 1) with eigenvalues 2 and
−l2.

After blow-down we obtain the local phase portraits of the origin of U2 which
depend on the signs of l1, l2 and l3. Doing all the combinations the origin of U2

consists of:

• two elliptic sectors and parabolic sectors, see phase portraits RPL.11 and
RPL.12 of Figure 6;
• two hyperbolic sectors and parabolic sectors, see phase portraits RPL.13

and RPL.14 of Figure 6;
• six hyperbolic sectors, see phase portrait RPL.15 of Figure 6.

Finally if we consider β = 1 and after eliminating the common factor v the origin
of the local chart U2 is either a hyperbolic node or a hyperbolic saddle, described
respectively by the phase portraits RPL.16 and RPL.17 of Figure 6. The Table 4
has the values of the parameters that realizes the phase portraits of Figure 6.

�

Proposition 7.8. Each real planar quadratic differential system with a double real
invariant straight line and a third invariant straight line having a Darboux invariant
can be written, after an affine change of coordinates, as system (DL), with γ = 0
and α 6= 0, and the Darboux invariant is

I15(x, y, t) = e−α tyx−β .



48 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Moreover there are 3 non-equivalent phase portraits in the Poincaré disc for this
systems. They are described by DL.1–DL.3 in Figure 6.

Proof. Let f1 = x = 0 be the double real invariant straight line. By the proof of
Proposition 4.3 we know that the second invariant straight line is f2 = y = 0. The
cofactors of f1 and f2 are, respectively, k1 = x, k2 = α + βx + γy. Equation (3.3)
with s ∈ R\{0} has only one solution γ = 0, s = −αλ2, λ1 = −βλ2. Taking λ2 = 1
and using this solution we obtain

ẋ = x2, ẏ = y(α+ βx),

with Darboux invariant I15(x, y, t) = e−α tyx−β .
To study the global phase portraits of systems (DL), since α 6= 0 we can take

α = 1. The origin of the system is the only finite singularity, which is a saddle-node.
For the infinity singularities we assume first that β − 1 6= 0. In the local chart U1

the origin is a saddle if β − 1 > 0, and a stable node if β − 1 < 0. In the chart U2

the system becomes

u̇ = −u((β − 1)u+ v), v̇ − v(βu+ v),

and the origin is a linearly zero singularity. Applying the blow up u = U , v = UW
we obtain the system

U̇ = −U2(β − 1 +W ), Ẇ = −UW,

which after eliminating the common factor U has the origin as only singular point.
If β − 1 > 0 the origin is a hyperbolic stable node and if β − 1 < 0 the origin is a
saddle.

After blow down we obtain the local phase portraits of the origin of U2 which
depend on β. When β − 1 > 0 the origin has two elliptic sectors and parabolic
sectors, see phase portrait DL.1 of Figure 6. If β − 1 < 0 then there are two
hyperbolic sectors and parabolic ones, see phase portrait DL.2 of Figure 6.

When β = 1 the infinity is filled up of singular points and in the local chart U2

the origin is a stable node. The phase portrait is described by DL.3 of Figure 6. �

Proposition 7.9. Each real planar quadratic differential system with two parallel
complex invariant straight lines and a third invariant straight line having a Darboux
invariant can be written, after an affine change of coordinates, as system (CPL).
A Darboux invariant is given by

I16(x, y, t) = etearctan(1/x)

Moreover there are 7 non-equivalent phase portraits in the Poincaré disc for this
system. They are described by CPL.1–CPL.7 in Figure 7.

Proof. Let f1 = x + i = 0, f2 = x − i = 0 be the two complex parallel straight
lines. By the proof of Proposition 4.3 we know that the third invariant straight
line is f3 = y = 0. The cofactors of f1, f2 and f3 are, respectively, k1 = x− i, k2 =
x + i, k3 = α + βx + γy. The equation (3.3) with s ∈ R \ {0} has two solutions,
namely

s1 = {γ = 0, s = i(2λ1 + (β + iα)λ3), λ2 = −βλ3 − λ1}
s2 = {s = 2iλ1, λ2 = −λ1, λ3 = 0}.



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 49

Using s2 (which is more general) we conclude that all systems with two parallel
complex straight lines and a real straight line as invariants curves have a Darboux
invariant. Moreover taking λ1 = −i/2 we obtain

I16(x, y, t) = et(x− i)i/2(x+ i)−i/2.

Using the polar form of the complex numbers it follows that (x− i)i/2(x+ i)−i/2 =
earctan(1/x) so the Darboux invariant is I16(x, y, t) = earctan(1/x)+t.

In [10] the authors already study the quadratic systems with f = x2+1 = 0 as an
invariant curve, given by ẋ = x2 + 1, ẏ = Q(x, y), with Q an arbitrary polynomial
of degree 2. In this paper we have Q(x, y) = y(α+βx+γy). So the system studied
here is a subcase of systems (VI) in [10]. In [10] the study of those systems is
divided in six cases and since we have the invariant straight line y = 0 there are
seven possible phase portraits. The case (VI.1) provides the phase portraits 1 and
2 of [10, Fig. 1], i.e. the phase portraits CPL.1 and CPL.2 of Figure 7; the case
(VI.2) gives the phase portrait 6 of [10, Fig. 1], i.e. the phase portrait CPL.3 of
Figure 7; the case (VI.4) generates the phase portraits 16 and 17 of [10, Fig 1],
i.e. the phase portraits CPL.4 and CPL.5 of Figure 7; the case (VI.5) gives the
phase portrait 20 of [10, Fig. 1], i.e. the phase portrait CPL.6 of Figure 7. Finally
the case (VI.6) provides the phase [10, portrait 21 Fig. 1], i.e. the phase portrait
CPL.7 of Figure 7. �

Proposition 7.10. Each real planar quadratic differential system with two complex
invariant straight lines that intersects in a real point and a third invariant straight
line having a Darboux can be written, after an affine change of coordinates, as one
of the following forms

(i) (p.1) with α3(β − 2β3) 6= 0 and Darboux invariant

I17(x, y, t) = eα3(β−2β3) te−2γ3 arctan(y/x)(x2 + y2)β3y−β .

(ii) (p.2) with c 6= 0, α = −1 and Darboux invariant

I18(x, y, t) = e− arctan(y/x)−ct

Moreover there are 5 non-equivalent phase portraits in the Poincaré disc for these
systems. They are described by p.1.1–p.1.3 and p.2.1, p.2.2 in Figure 7.

Proof. Let f1 = x + iy = 0 and f2 = x− iy = 0 be the two complex straight lines
that intersect at a real point. We have two systems, (p.1), with f3 = y, and (p.2)
with f3 = y + ax + c. We shall do the calculations for (p.1), and for system (p.2)
the computations are analogous.

we consider system (p.1) the cofactors of f1, f2 and f3 are, respectively,

k1 = (1/2)(βx+ 2γ3y + 2α3 − i(β − 2β3)y),

k2 = (1/2)(βx+ 2γ3y + 2α3 + i(β − 2β3)y),

k3 = α3 + β3x+ γ3y.

Solving equation (3.3) the most general solution is

λ1 = β3 + iγ3, λ2 = β3 − iγ3, λ3 = −β, s = α3(β − 2β3).

Hence assuming α3(β − 2β3) 6= 0 system (p.1) of Theorem 2.3 has the Darboux
invariant

I17(x, y, t) = eα3(β−2β3) ty−β(x− iy)β3−iγ3(x+ iy)β3+iγ3 .



50 J. LLIBRE, R. D. S. OLIVEIRA, C. A. B. RODRIGUES EJDE-2021/69

Using the polar form of the complex numbers it follows that

(x− iy)β3−iγ3(x+ iy)β3+iγ3 = e−2γ3 arctan(y/x) (x2 + y2)β3

and we obtain the Darboux invariant

I17(x, y, t) = eα3(β−2β3)te−2γ3 arctan(y/x) (x2 + y2)β3y−β

For system (p.2) the third invariant straight line is f3 = y + ax+ c with c 6= 0. In
this case the system has a Darboux invariant if and only if α = −1, and with the
same reasoning applied above we obtain the invariant

I18(x, y, t) = e− arctan(y/x)−c t.

We start the study of the global phase portraits with systems (p.1). Since α3 6= 0
we can take α3 = 1. Systems (p.1) have at most two finite singularities, namely
z1 = (0, 0) and z2 = (−2/β, 0). When β = 0 the point z2 goes to infinity. The point
z1 is an unstable node and the eigenvalues associated to z2 are −1 and (β−2β3)/β.
So the point z2 is either a stable node or a saddle.

In the local chart U2 the origin is not a singularity for the compactfied system.
In the local chart U1 the system compactified has only one infinity singularity
u1 = (0, 0) with eigenvalues −β/2 and −(β − 2β3)/2.

Then if β(β − 2β3) > 0, z2 is a saddle and u1 is a stable node and the only
phase portrait is p.1.1 of Figure 7, realizable for β = 1, γ3 = 1 and β3 = −1/2. If
β(β − 2β3) < 0, z2 is a stable node and u1 is a saddle and the corresponding phase
portrait of this case is p.1.2 of Figure 7, realizable for β = 1, γ3 = 1 and β3 = 3/2.
Finally if β = 0 then z2 goes to the infinity and u2 becomes a semi hyperbolic
saddle-node generating the phase portrait p.1.3 of Figure 7, which is realizable for
β = 0, γ3 = 1 and β3 = 2.

To study the global phase portraits of systems (p.2) we start with the infinity
singular points. In the local chart U1 system (p.2) becomes

u̇ = −cv(u2 + 1), v̇ = −v(aβ + au+ cuv + βcv + βu− 1).

So the line v = 0 is filled up of singular points. The same happens in the local
chart U2. In the finite part the point (0, 0) is the only singularity, with complex
eigenvalues. So the origin can be a node or a center. Both cases are described,
respectively, by the phase portraits p.2.1, realizable with a = β = 1 and c = 2, and
p.2.2, realizable with a = 1, β = 0 and c = 2, of Figure 7. �

Summarizing the nine propositions about quadratic systems with a invariant
reducible cubic and a Darboux invariant, we present table 5. In this table we
expose the relation between the normal forms and the phase portraits that can
occur, as well as the Figure where the corresponding phase portrait is given in this
manuscript.

By the end we prove Theorem 2.5. This result is about the differential systems
having an invariant cubic but that do not have a Darboux invariant.

Proof of Theorem 2.5. First we consider systems of type (CE), i.e, the ones which
has an invariant cubic of the form f = f1f2 = 0 where f1 = x2 + y2 + 1 and
f2 = ax+ by + c. By Theorem 2.3 these systems can be written as

ẋ = −(x2 + y2 + 1)− 2α1y(y + ax+ c), ẏ = a(x2 + y2 + 1) + 2α1x(y + ax+ c),

with f1 = x2 + y2 + 1 and f2 = y + ax + c. The cofactors of f1 and f2 are
k1(x, y) = 2(ay + x) and k2(x, y) = −2α1(ay − x), respectively. So the cofactors



EJDE-2021/69 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 51

Table 5. Table of relations among all the normal forms and the
possible phase portraits of systems which have a Darboux invari-
ant.

Normal form Cond. for a Darboux invariant Possible phase portratis
(E.2) c = −1, β1 = 2β2, α2 6= 0 EL.2.1–EL.2.2 (Figure 2)
(H.2) c = −1, γ1 = 2γ2, α2 6= 0 HL.2.1–HL.2.3 (Figure 2)
(H.3) c = 0, β = −γ, Aα 6= 0 HL.3.1–HL.3.5 (Figure 2)
(H.3) β = γ = 0, α 6= 0 HL.3.6–HL.3.9 (Figure 2)
(H.4) A = 2β, α 6= 0 HL.2.1–HL.2.3 (Figure 2)
(P.1) α1 − 2α2 6= 0 PL.1.1–PL.1.24 (Figure 3)

PL.1.25–PL.1.30 (Figure 4)
(P.2) β1 = β2, α2 = 0, cγ2 6= 0 PL.2.1–PL.2.4 (Figure 4)
(P.2) c = γ2 = 0, α2(β2 − β2) 6= 0 PL.2.5–PL.2.11 (Figure 4)

(LV.1) α(q − β) 6= 0 LVL.1.1–LVL.1.6 (Figure 5)
(LV.2) c = −1 LVL.2.1–LVL.2.17 (Figure 5)
(LV.2) c = q = 0, p 6= 0 DL.1–DL.3 (Figure 6)
(LV.3) α = −(β + 1), cβ 6= 0 LVL.3.1 (Figure 5)
(RPL) always has a Darboux invariant RPL.1–RPL.17 (Figure 6)
(DL) γ = 0, α 6= 0 DL.1–DL.3 (Figure 6)

(CPL) always has a Darboux invariant CPL.1–CPL.7 (Figure 7)
(p.1) α3(β − 2β3) 6= 0 p.1.1–p.1.3 (Figure 7)
(p.2) α = −1, c 6= 0 p.2.1–p.2.2 (Figure 7)

have no constant terms, i.e., k1(0, 0) = k2(0, 0) = 0. The consequence of this is
that equation (3.3) has no solution considering s 6= 0. Hence these systems do not

have a Darboux invariant of the form estfλ1
1 fλ2

2 .
The proofs for the other systems are very similar. In fact it suffices to observe

that the cofactors of the invariant curves never have a constant term. �

Acknowledgements. J. Llibre was supported by the Ministerio de Ciencia, Inno-
vación y Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-
I00, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617,
and the H2020 European Research Council grant MSCA-RISE-2017-777911. R.
Oliveira was supported by FAPESP “Projeto Temático” grant 2019/21181-0 and
by the CNPq grant 304766/2019-4 (Produtividade em Pesquisa). We want thank
to the referees for their comments and suggestions which help us to improve the
presentation of this article.

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Jaume Llibre
Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,

Barcelona, Catalonia, Spain

Email address: jllibre@mat.uab.cat

Regilene Oliveira

Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida

Trabalhador São-carlense, 400, 13.560-970, São Carlos, SP, Brazil
Email address: regilene@icmc.usp.br

Camila A. B. Rodrigues
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Flo-

rianópolis, Santa Catarina, Brazil

Email address: c.r.lima@ufsc.br


	1. Introduction and statements of results
	2. Statement of the main results
	2.1. Reducible invariant cubics

	3. Preliminary and basic results
	3.1. Invariants
	3.2. Darboux invariants

	4. Poincaré compactification
	4.1. Irreducible invariant cubics
	4.2. Reducible invariant cubics

	5. Proof of Theorems ?? and ??
	6. Proof of Theorem ??
	6.1. Systems of type (E)
	6.2. Systems of type (CE)
	6.3. Systems of type (H)
	6.4. Systems of type (P)
	6.5. Systems of type (LV)
	6.6. Systems of type (RPL)
	6.7. Systems of type (DL)
	6.8. Systems of type (CPL)
	6.9. Systems of type (p)

	7. Proof of Theorems ?? and ??
	Acknowledgements

	References