Adv Syst Sci Appl 2025; 02:92–104
Published online at https://ijassa.ipu.ru.

Vector Fields with Non-Isolated Singular Points
Viewed from the Inside and Outside

Natalia Pavlova1,2 Alexey Remizov1*

1Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia
2RUDN University, Moscow, Russia

Abstract: We present a brief survey of recent results about vector fields with non-isolated
singular points and their applications. One of the most interesting and promising application is
connected with quasi-linear differential equations of the second order, including the equation of
geodesics in signature varying (pseudo-Riemannian) metrics.

Keywords: vector fields, singular points, center manifold, normal forms, resonances

1. INTRODUCTION

We start with a general construction, which naturally leads us to vector field with non-isplated
singular points (more precisely, singular points fill a manifols of codimension two in the phase
space).

Let M be a real smooth (C∞) manifold of dimension n+ 2. Here and further we use the
following standard notations:

• C∞(M) is the ring of smooth functions on M

• Γ(TM) is the module of smooth vector fields on M

• Γ(T ∗M) is the module of smooth covector fields (1-forms) on M

Consider a distribution on M defined via n+ 1 differential 1-forms:

ω1 = 0, . . . , ωn+1 = 0, ωi ∈ Γ(T ∗M). (1.1)

At points of M where the 1-forms ω1, . . . , ωn+1 are linearly independent, system (1.1)
defines a one-dimensional distribution, that is, a direction field, which can be a class of
collinear vector fields, such points we shall call regular. At points of M where the 1-forms
ω1, . . . , ωn+1 are nor linearly independent, the distribution has dimension greater then 1, such
points we shall call singular.

In (local) coordinates u = (u1, . . . , un+2) on M system (1.1) yields the Pfaffian system

ωi =
n+2∑
j=1

ωij(u) duj = 0, i = 1, . . . , n+ 1, (1.2)

with (n+ 1)× (n+ 2) matrix Ω = (ωij).

∗Corresponding author: alexey-remizov@yandex.ru



VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 93

Singular points fill a stratified manifold Σ ⊂ M that consists of the strata

Σi = {u ∈ M : rg Ω(u) = (n+ 1)− i}, i = 1, . . . , n+ 1.

The dimension of Σi decreases rapidly with increasing i, namely: codimΣi = i(i+ 1); see,
for example, [2]. The maximal stratum Σ1 of codimension 2 is stable with respect to small
perturbation of ωi, while the strata of higher codimension are, generally speaking, not.

The goal of the paper is to study integral curve of the distribution (1.1) entering its singular
points of the maximal stratum Σ1. We also consider some application of the obtained results
for studying singularities of differential equations of special types, which are interesting due
to various applications.

2. THE MAIN RESULTS

The distribution (1.2) can be determined (at least locally) by a smooth vector field V̄ ∈
Γ(TM) that is zero at points of Σ. This field is unique up to multiplying by a scalar factor:

V̄ =
n+2∑
j=1

vj(u)
∂

∂uj

, vj = (−1)j+1∆j, j = 1, . . . , n+ 2, (2.3)

where ∆j is the minor of Ω obtained by elimination of the j-th column.
Example 2.1:
Consider the lowest dimension case: n = 1. Then the distribution (1.2) in 3-dimensional
space is the intersection of two fields of planes:

ωi = ωi1du1 + ωi2du2 + ωi3du3 = 0, i = 1, 2.

Here Ω is a 2× 3 matrix

Ω =

 ω11 ω12 ω13

ω21 ω22 ω23


and V̄ is the vector product of its columns:

v1 =

∣∣∣∣∣∣ω12 ω13

ω22 ω23

∣∣∣∣∣∣ , v2 = −

∣∣∣∣∣∣ω11 ω13

ω21 ω23

∣∣∣∣∣∣ , v3 =

∣∣∣∣∣∣ω11 ω12

ω21 ω22

∣∣∣∣∣∣ .
The maximal stratum Σ1 is defined by the condition rg Ω = 1, that is, the matrix Ω has at

least one non-zero element. Without loss of generality, assume that ωi3 ̸= 0. Then the equality
ωi1v1 + ωi2v2 + ωi3v3 = 0 yields the expression

v3 = −ωi1

ωi3

v1 −
ωi2

ωi3

v2, (2.4)

which is valid locally, in a neighborhood of a point where ωi3 ̸= 0.
Expression (2.4) shows that the components of the vector field V̄ are connected by

functional relations: in a neighborhood of a point of Σ1 they belong to the ideal in the ring
C∞(M) generated by two of them.

The letter property established for n = 1, is valid for arbitrary n:

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94 N. G. PAVLOVA, A. O. REMIZOV

Lemma 2.1:
In a neighborhood of a point of Σ1, all components of the field (2.3) belong to the ideal
(in C∞(M)) generated by two of them. Therefore, one can choose coordinates x, y, z =
(z1, . . . , zn) such that the germ of the field (2.3) has the form

ẋ = v, ẏ = w, żi = aiv + biw, i = 1, . . . , n, (2.5)

where v, w, ai, bi ∈ C∞(M).

Proof
The proof of the first statement of the lemma is similar to those for the case n = 1 considered
above. The second statement obviously follows from the first one.

Lemma 2.1 establish some important properties of vector fields defined by Pfaffian
systems (1.2). The main distinctive feature is that singular points of such fields are not
isolated, but filled a manifold of codimension two. For instance, singular points of the field
(2.5) are given by two equations

v(x, y, z) = 0, w(x, y, z) = 0. (2.6)

Therefore, the spectrum of the linear part of field (2.5) at every singular points T ∈ Σ1

has the form
spec (T ) = (λ1(T ), λ2(T ), 0, . . . , 0), #0 = n,

where the eigenvalues λ1,2(T ) continuously depend on T ∈ Σ1. Further we shall use the
notation

λ1,2 = λ1,2(T0).

Further we shall consider the germ (2.3) at a generic singular point T0 ∈ Σ1, where

Reλ1,2 ̸= 0. (2.7)

In this case, the set of singular points Σ1 is the (unique) center manifold W c of the field (2.5)
passing through the point T0. This allows us to apply the reduction principle [1, 6], which
yields the following result:
Theorem 2.1:
The germ of the field (2.5) at a generic singular point T0 satisfying the condition (2.7) is
topologically equivalent to the field

ξ̇ = a1ξ, η̇ = a2η, ζ̇i = 0, i = 1, . . . , n, (2.8)

where aj = sgn(Reλj).

Proof
This is a trivial corollary of the reduction principle, which states that the germ of the field
(2.5) at T0 is topologically equivalent to the product of the field ξ̇ = a1ξ, η̇ = a2η and the
restriction of (2.5) to its center manifold W c. Since W c consists of singular points of the field
(2.5), this yields normal form (2.8).

The next step in the study of fields (2.5) is their smooth local classification. For this,
we need to consider two types of resonances (integer relations) between the non-zero
eigenvalues:

s1λ1 + s2λ2 = 0, si ∈ Z+, i = 1, 2, (2.9)
s1λ1 + s2λ2 = λj, si ∈ Z+, i, j = 1, 2. (2.10)

From them we have to exclude trivial resonances, which always exist: resonance (2.9) with
s1 = s2 = 0 and resonance (2.10) with s1 = 1, s2 = 0, j = 1 or s1 = 0, s2 = 1, j = 2.

The number |s| = s1 + s2 is called the order of resonance (2.9) or (2.10).

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VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 95

Remark 2.1:
1. The absence of resonances (2.10) implies the absence of the resonances (2.9):

2. In the absence of resonances (2.9), resonances (2.10) may have only the simplest form

λ1 = mλ2 or λ2 = mλ1 (2.11)

with integer m ≥ 1.
Theorem 2.2:
If between the eigenvalues λ1,2(T ) there are no non-trivial resonances (2.9) of any order
|s| ≥ 1 for all T ∈ W c sufficiently close to T0, then the germ of the field (2.5) at T0 is C∞-
smoothly equivalent to

ξ̇ = X(ξ, η, ζ), η̇ = Y (ξ, η, ζ), ζ̇j = 0, j = 1, . . . , n, (2.12)

where X and Y are smooth functions vanishing on the center manifold.
If, in addition, |λ1| > |λ2| at T0, then

X = λ1(ζ)ξ + φ(ζ)ηm, Y = λ2(ζ)η, (2.13)

where φ(ζ) ̸≡ 0 only if λ1 = mλ2 with some integer m > 1.
The condition that between λ1,2(T ) there are no non-trivial resonances (2.9) of any order

|s| ≥ 1 for all T ∈ W c appears when we consider infinitely smooth classification of vector
fields with non-isolated singular points. First it was formulated in the paper [17] and then it
is often named after him.
Remark 2.2:
1. If the pair λ1,2 = λ1,2(T0) belongs to the Poincaré domain, that is, λ1,2 are real and of the
same sign or complex conjugate with the condition (2.7), then the Roussarie condition holds
true, whence Theorem 2.2 is valid.

2. If the pair λ1,2 = λ1,2(T0) belongs to the Siegel domain, that is, λ1,2 are real and of
different signs, the Roussarie condition holds true if and only if

λ1(T ) : λ2(T ) ≡ const /∈ Q, ∀T ∈ W c.

If we consider Ck-smooth equivalence with k < ∞, the absence of resonances of all
orders |s| at all points T ∈ W c can be replaced with the absence of resonances of the orders
|s| ≤ N(k) at T0, where

N(k) = 2

[
(2k + 1)

max |Reλ1,2|
min |Reλ1,2|

]
+ 2, (2.14)

the square brackets denote the integer part of a number. The estimation (2.14) is taken
from [18].
Theorem 2.3:
If between the eigenvalues λ1,2 = λ1,2(T0) there are no non-trivial resonances (2.9) of any
order 1 ≤ |s| ≤ N(k), then the germ of the field (2.5) at T0 is Ck-smoothly equivalent to

ξ̇ = X(ξ, η, ζ), η̇ = Y (ξ, η, ζ), ζ̇j = 0, j = 1, . . . , n,

where X and Y are smooth functions vanishing on the center manifold.
If, in addition, |λ1| > |λ2| at T0, then

X = λ1(ζ)ξ + φ(ζ)ηm, Y = λ2(ζ)η,

where φ(ζ) ̸≡ 0 only if λ1 = mλ2 with positive integer m ≤ N(k).

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96 N. G. PAVLOVA, A. O. REMIZOV

For the proofs of Theorems 2.2, 2.3, see [5, 13].

Remark 2.3:
Geometrically, Theorems 2.2, 2.3 state that the vector field (2.5) has 2-dimensional invariant
foliation such that the restriction of the field to its leaves has the spectrum λ1,2(T ), that is, it
is a node or a saddle od a focus. See, for example, Fig. 2.1.

From the analytical viewpoint, Theorems 2.2, 2.3 are generalization of the Poincare –
Dulac normal form for vector fields that have zero eigenvalues.

Fig. 2.1. Illustration of Theorems 2.1 – 2.3: local phase portraits of the vector field (2.5). Here n = 1 and W c

coincides with the z-axis.

3. ROUSSARIE VECTOR FIELDS

Now we consider vector fields of the form (2.5) satisfying the additional condition:

spec (T ) ≡ (λ1(T ), λ2(T ), 0, . . . , 0), ∀T ∈ W c,

where λ1,2(T ) are non-zero real numbers such that

qλ1(T ) + pλ2(T ) = 0, p, q ∈ Z+, gcd (p, q) = 1, ∀T ∈ W c. (3.15)

Such vector fields are named after Robert Roussarie, who studied the partial case

p = q = 1 (3.16)

in his [17]. This study was motivated by the degeneracy of closed differential 2-forms.
A generic closed 2-form ω on the 4-dimensional real space degenerates on a smooth 3-
dimensional manifold Σ. At a generic point of Σ, the 2-dimensional kernel of the form ω
is transversal to Σ, and the germ of ω can be reduced to

p1dp1 ∧ dq1 + dp2 ∧ dq2.

However, the kernel of ω is tangent to Σ at some points, which generically fill a curve S ⊂ Σ,
the normal form of ω at points of S is more complicated. The kernel of ω cuts out a direction
field on Σ, which can be given (uniquely up to a scalar factor) by a vector field of the form
(2.5), whose singular points fill the curve S. The condition that ω is closed implies that the
trace of the linear part of the vector field at every its singular point is zero, that is, the non-zero
eigenvalues λ1,2 satisfy the resonance (3.15) with p = q = 1.

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VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 97

Actually, in many problems we are interested not in vector fields themselves, but in the
corresponding direction fields. Therefore, when bringing a vector field to its normal forms,
one can multiply it by a non-vanishing scalar function in addition to changes of the phase
variables. Normal forms thus obtained are called orbital. Orbital normal forms allow us to
get rid of a redundant module, and therefore, to simplify the classification.

Theorem 3.1:
The germ of every Roussarie vector field is C∞-smoothly orbitally equivalent to

ẋ = px(1 + Φ1(r, z)), ẏ = qy(−1 + Φ2(r, z)),

żi = rΨi(r, z), i = 1, . . . , n,
(3.17)

where r = xqyp is the resonant monomial of the resonance (3.15), Φ1,2,Ψi ∈ C∞(M) and
Φ1(0, 0) = Φ2(0, 0) = 0.

Theorem 3.1 establishes a generalization of the Poincare – Dulac normal form for
Roussarie vector fields: the functions Φ1,2 and Ψi contain all resonant terms. It is well-
known that the Poincare – Dulac normal form allows further simplification. To obtain such
simplification for (3.17), we shall use the notion of the quotient vector field.

The field (3.17) generates the field in the (r, z)-space:

ṙ = ˙(xqyp) = qxq−1ypẋ+ pxqyp−1ẏ =

qxq−1yppx(1 + Φ1) + pxqyp−1qy(−1 + Φ2) = pqr(Φ1 + Φ2),

żi = rΨi(r, z), i = 1, . . . , n.

Reducing the common factor r, we get the quotient vector field for (3.17):

ṙ = pqΦ(r, z), żi = Ψi(r, z), i = 1, . . . , n, (3.18)

where
Φ(r, z) = Φ1(r, z) + Φ2(r, z).

Remark 3.1:
From the resonant relation (3.15) it follows that Φ(0, z) ≡ 0 for all z, i.e., the restriction of
Φ(r, z) to the center manifold is identically zero.

Generically, for almost all points T0 ∈ W c the field (3.17) satisfies the following
condition:

∃i ∈ {1, . . . , n} : Ψi(0, 0) ̸= 0. (3.19)

Equivalently, 0 is not a singular point of its quotient field (3.18). This allows us to establish
the existence of n independent first integrals of the field (3.18), which are obviously first
integrals of the field (3.17). For details, see [5, 13].

Using these first integrals, one can prove the following theorem:

Theorem 3.2:
If the condition (3.19) for (3.17) holds true, the germ of the Roussarie vector field is C∞-
smoothly orbitally equivalent to

ẋ = px, ẏ = −qy, ż1 = xqyp, żi = 0, i = 2, . . . , n. (3.20)

Moreover, it is Ck−1-smoothly orbitally equivalent to

ẋ = px, ẏ = −qy, żi = 0, i = 1, . . . , n, (3.21)

where k = max{p, q}, but in general it is not Ck-smoothly equivalent.

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98 N. G. PAVLOVA, A. O. REMIZOV

Fig. 3.2. Two examples of C0 saddle surfaces of the vector field ẋ = x, ẏ = −y, ż = xy: z = − 1
2xy ln |y/x|

(left) and z = −xy ln |y| (right).

Proof
The first statement of the theorem is proved in the case p = q = 1 by Roussarie in 1975. Other
statements are proved in [10].

Example 3.1:
Consider the vector field

ẋ = x, ẏ = −y, ż = xy, (x, y, z) ∈ R3.

Any saddle surface of this field has the form z = −1
2
F (x, y), where

F (x, y) = f(xy) + xy ln
∣∣∣y
x

∣∣∣, if xy ̸= 0,

and F (x, y) = 0, if xy = 0. Here f is an arbitrary continuous function.
The obtained formula shows that F is continuous (for continuous f ), but it is never C1.

See examples in Fig. 3.2.

4. APPLICATIONS: QUASI-LINEAR ODES OF THE SECOND ORDER

Consider the differential equation

∆(x, y)
dp

dx
= M(x, y, p), p = dy/dx, (4.22)

where ∆(x, y), M(x, y, p) are smooth functions, M is analytic in p. Generically, the set of
singular points of equation (4.22) is a regular curve

Γ = {(x, y) : ∆(x, y) = 0}.

If the point q0 = (x0, y0) /∈ Γ, then for every direction p0 equation (4.22) has a unique
solution satisfying the initial condition

y(x0) = y0, p(x0) = p0.

The situation is more complicated if q0 = (x0, y0) ∈ Γ. For example, there may be solutions
whose oscillations accumulate near a singular point.

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VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 99

Example 4.1:
The equation x4dp/dx = 2x3p− (2x2 + 1)y has a family of solutions

y(x) = x2(α cosx−1 + β sinx−1), α, β = const,

Except for α = β = 0, all solutions are oscillating at x = 0:

∃ lim
x→0

y(x) = 0, ̸ ∃ lim
x→x0

y′(x) (but y′(0) = 0).

Example 4.2:
The equation x2dp/dx = xp− 2y has a family of solutions

y = x(α cos ln |x|+ β sin ln |x|), α, β = const,

Except for α = β = 0, all solutions are oscillating at x = 0:

∃ lim
x→0

y(x) = 0, ̸ ∃ lim
x→x0

y′(x) (and ̸ ∃ y′(0)).

The above examples motivate the following formal definition:
Definition 4.1:
Oscillating solutions entering q0 are solutions such that

∃ lim
x→x0

y(x) = y0, ̸ ∃ lim
x→x0

p(x).

Here x → x0 may be either two-sided or one-sided limit.
The above examples show that oscillating solutions exist. However, the following theorem

states that oscillating solutions do not exist generically.
Theorem 4.1:
Let q0 ∈ Γ and M(q0, p) is an analytic function not identically zero. Then equation (4.22) has
no oscillating solutions entering the point q0. Moreover, solutions can enter q0 at so-called
admissible directions p that correspond to real roots of M(q0, p) only.

The proof of Theorem 4.1 is based on the analysis of the distribution defined by equation
(4.22) in the (x, y, p)-space:

ω1 = ∆dp−Mdx = 0, ω2 = dy − pdx = 0,

which generates the vector field

ẋ = ∆(x, y), ẏ = p∆(x, y), ṗ = M(x, y, p). (4.23)

All components of the field (4.23) belong to the ideal generated by ∆ and M . Given q0 ∈ Γ,
the admissible directions p correspond to singular points (q0, p) of the vector field (4.23)
given by two equations:

∆(q0) = 0, M(q0, p) = 0.

For more details about oscillating solutions see [11].
Further, we shall analyse vector filed (4.23) in order to study non-oscillating solutions of

equation (4.22) entering its singular points. First, remark that vector filed (4.23) has the form
(2.5) with n = 1, where the variables x, p in (4.23) correspond to x, y in (2.5), the variable y
in (4.23) corresponds to z1 in (2.5).

Let q0 ∈ Γ and M(q0, p∗) = 0, i.e., p∗ is an admissible direction at q0. The spectrum of
the linear part of the field (4.23) at singular point (q0, p∗) is

spec (q0, p∗) = (λ1, λ2, 0),

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100 N. G. PAVLOVA, A. O. REMIZOV

where the eigenvalues

λ1 = (∆x + p∆y)(q0, p∗), λ2 = Mp(q0, p∗).

Assume that λ1,2 ̸= 0. Then we define the value λ = λ2 : λ1, which determines the
number and the behavior of solutions of equation (4.22) that enter q0 with the direction p∗.
Namely, the following result is obtained in [15].
Theorem 4.2:
If λ < 0, then equation (4.22) has only one solutions passing though the point q0 with the
tangential direction p∗.

If λ > 0, then equation (4.22) has an infinite number of solutions entering the point q0 with
the direction p∗. In appropriate local coordinates, these solutions have one of two following
forms:

y = F (x, c|x|λ), if λ /∈ N,
y = F (x, xλ(c+ ε ln |x|)), ε ∈ {0, 1}, if λ ∈ N,

where F is a smooth function, c = const.

Fig. 4.3. Illustration for Theorem 4.2: integral curves of the field (4.23) and their projections to the (x, y)-plane
– solutions of equation (4.22). From left to right: λ < 0 (a), 0 < λ < 1 (b), λ > 1 (c). The curve Γ is depicted

as a dashed curve on the (x, y)-plane.

4.1. Quasi-Linear ODEs of the Second Order Cubic in p

Consider an important class of equations (4.22), where M(x, y, p) is a cubic polynomial in p:

M = µ0 + µ1p+ µ2p
2 + µ3p

3, µi = µi(x, y).

An attention to such equations is motivated by their role in physics and geometry, for
instance, the description of various geometric structures (geodesic flows in affine or projective
connection, etc.). Equations of this class were studies by Sophus Lie, A. Tresse, J. Liouville,
E. Cartan, etc. See also the recent papers [7, 19, 20].

For a generic cubic polynomial M(q0, p) and almost all points q0 ∈ Γ there are 4 possible
cases (the abbreviation AD below means admissible direction):

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VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 101

• 1 AD p0 with λ(q0, p0) > 0;
• 3 ADs p0, p1, p2 with λ(q0, p0) > 0 and λ(q0, pi) < 0, i = 1, 2;
• 1 AD p0 with λ(q0, p0) < 0;
• 3 ADs p0, p1, p2 with λ(q0, p0) < 0 and λ(q0, pi) > 0, i = 1, 2.

Example 4.3:
Consider the equation

x
dp

dx
= αp(p2 − 1), α ̸= 0, (4.24)

whose singular points fill the curve Γ = {x = 0}. For every point q0 ∈ Γ the cubic polynomial

M(p) = αp(p2 − 1)

has three different real roots: p0 = 0 and p1,2 = ±1, where λ(q0, 0) = −α and λ(q0,±1) =
2α. Solution of equation (4.24) entering the origin are presented in Fig. 4.4.

Fig. 4.4. On the left: the case α > 0, on the right: the case α < 0. The bold lines are solutions that are unique
with given AD.

Example 4.4:
Consider the equation

x
dp

dx
= αp(p2 + 1), α ̸= 0, (4.25)

whose singular points fill the curve Γ = {x = 0}. For every point q0 ∈ Γ the cubic polynomial

M(p) = αp(p2 + 1)

has one real root p0 = 0 and λ(q0, 0) = α.

Fig. 4.5. On the left: the case α > 0, on the right: the case α < 0. The bold lines are solutions that are unique
with given AD.

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102 N. G. PAVLOVA, A. O. REMIZOV

4.2. Equation of Geodesics in Signature Varying Metrics
Consider a pseudo-Riemannian metric

ds2 = a(x, y) dx2 + 2b(x, y) dxdy + c(x, y) dy2,

where a, b, c ∈ C∞(M) and the discriminant function

∆ = ac− b2

vanishes and changes its sign on the regular curve Γ ⊂ M . The latter condition implies that
the coefficients a, b, c do not vanish simultaneously, therefore, the square polynomial

a(q0) + 2b(q0)p+ c(q0)p
2

at every point q0 ∈ Gamma has the double root

p0 = −a

b
= −b

c
,

which defines the isotropic direction of the metric at the point q0.
The equation of (unparametrized) geodesics in this metric has the form

∆(x, y)
dp

dx
=

1

2
M(x, y, p), p = dy/dx,

where M is a cubic polynomial

M = µ0 + µ1p+ µ2p
2 + µ3p

3

with the coefficients
µ3 = c(2by − cx)− bcy,

µ2 = b(2by − 3cx) + 2ayc− acy,

µ1 = b(3ay − 2bx) + axc− 2acx,

µ0 = a(ay − 2bx) + axb.

This is a special case of quasi-linear equation of the second order.
In this case, the isotropic direction p0 is always admissible, that is, M(q0, p0) = 0. At

almost all points of the curve Γ, the cubic polynomial M has either one or three real roots,
and there are only two possible combinations:

• one AD p0 with λ(q0, p0) =
1
2
,

• three ADs p0, p1, p2 with λ(q0, p0) =
1
2

and λ(q0, pi) = −1, i = 1, 2.

A deep explanation of this surprising fact can be found in [5].
Finally, we remark one more interesting property established recently. Let q0 ∈ Γ and the

polynomial M at q0 has three different real roots p0, p1, p2, that is, three different admissible
directions at q0. Let pΓ be the tangent direction to the curve Γ at q0. Then the relative position
of the directions p0, p1, p2, pΓ is determined by their cross-ratio:

DV(p0, p1, pΓ, p2) = 2.

The proof can be found in [12].
In Fig. 4.6, we present three families of geodesics issuing from a point q0 ∈ Γ. Here the

curve Γ coincides with the horizontal axis (dashed line) and the isotropic admissible direction
p0 at all points q0 ∈ Γ is vertical. The families on the left and in the center correspond to points
q0 with unique admissible direction p0. The family on the right corresponds to a point q0 with
three admissible direction p0, p1, p2; geodesics with non-isotropic tangential directions p1, p2
are depicted as bold lines.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



VECTOR FIELDS WITH NON-ISOLATED SINGULAR POINTS... 103

Fig. 4.6. Three families of geodesics issuing from a point q0 ∈ Γ.

5. CONCLUSION

We presented a survey of recent results about vector fields with non-isolated singular points
and some their applications. One of the most interesting and promising application is
connected with quasi-linear differential equations of the second order, including the equation
of geodesics in signature varying (pseudo-Riemannian) metrics. This subject motivates
research in various directions, see, for example, the papers [9,14,16]. Applications of different
types can be also found in [3, 4], [7, 19], and [8].

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104 N. G. PAVLOVA, A. O. REMIZOV

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Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)


	Introduction
	The main results
	Roussarie vector fields
	Applications: Quasi-Linear ODEs of the Second Order
	Quasi-Linear ODEs of the Second Order Cubic in p
	Equation of Geodesics in Signature Varying Metrics

	Conclusion