Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena
In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type ⁽¹⁾ₗ, = 4, 5, 6, …. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group ˜(⁽¹⁾₅) and which we call the Sasano...
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| description | In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type ⁽¹⁾ₗ, = 4, 5, 6, …. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group ˜(⁽¹⁾₅) and which we call the Sasano system of type ⁽¹⁾₅. We prove that one family of the Sasano system of type ⁽¹⁾₅ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.
|
| first_indexed | 2026-03-21T11:56:45Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 020, 24 pages
Non-Integrability of the Sasano System of Type D
(1)
5
and Stokes Phenomena
Tsvetana STOYANOVA
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”,
5 J. Bourchier Blvd., Sofia 1164, Bulgaria
E-mail: cveti@fmi.uni-sofia.bg
Received November 28, 2023, in final form March 10, 2025; Published online March 27, 2025
https://doi.org/10.3842/SIGMA.2025.020
Abstract. In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine
Weyl group symmetry of type D
(1)
l , l = 4, 5, 6, . . . . In this paper, we study the integrability
of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl
group W̃
(
D
(1)
5
)
and which we call the Sasano system of type D
(1)
5 . We prove that one family
of the Sasano system of type D
(1)
5 is not integrable by rational first integrals. We describe
Stokes phenomena relative to a subsystem of the second normal variational equations. This
approach allows us to compute in an explicit way the corresponding differential Galois group
and therefore to determine whether the connected component of its unit element is not
Abelian. Applying the Morales–Ramis–Simó theory, we establish a non-integrable result.
Key words: Sasano systems; non-integrability of Hamiltonian systems; differential Galois
theory; Stokes phenomenon
2020 Mathematics Subject Classification: 34M55; 37J30; 34M40; 37J65
1 Introduction
In 2006, using the methods from algebraic geometry Y. Sasano [27] introduced higher-order
Painlevé systems, which have symmetry under the affine Weyl group of type D
(1)
l , l = 4, 5, 6, . . . .
Subsequently Fuji and Suzuki [13] derived the higher-order Painlevé systems of type D
(1)
2n+2 from
the Drinfeld–Sokolov hierarchy by similarity reduction. These higher-order Painlevé systems
have four essential properties:
1. They are Hamiltonian systems.
2. They admit an affine Weyl group symmetry of type D
(1)
l as Bäcklund transformations.
3. They can be considered as higher-order analogues of the Painlevé V and Painlevé VI
systems.
4. They have several symplectic coordinate systems, on which the Hamiltonians are polyno-
mial.
In this paper, we study the integrability of the following fourth-order Painlevé system
ẋ =
2x2y
t
+ x2 − 2xy
t
−
(
1 +
β
t
)
x+
α2 + α5
t
+
2z((z − 1)w + α3)
t
,
ẏ = −2xy2
t
+
y2
t
− 2xy +
(
1 +
β
t
)
y − α1,
ż =
2z2w
t
+ z2 − 2zw
t
−
(
1 +
α5 + α4
t
)
z +
α5
t
+
2yz(z − 1)
t
,
mailto:cveti@fmi.uni-sofia.bg
https://doi.org/10.3842/SIGMA.2025.020
2 Ts. Stoyanova
ẇ = −2zw2
t
+
w2
t
− 2zw +
(
1 +
α5 + α4
t
)
w − α3 −
2y(−w + 2zw + α3)
t
(1.1)
with β = 2α2 + 2α3 + α4 + α5, where α0, α1, . . . , α5 are complex parameters, which satisfy the
relation
α0 + α1 + 2α2 + 2α3 + α4 + α5 = 1.
The system (1.1) is one of the systems introduced by Sasano in [27]. It admits the extended affine
Weyl group W̃
(
D
(1)
5
)
as a group of Bäcklund transformations. For these reasons, throughout
this paper we call the system (1.1) the Sasano system of type D
(1)
5 or in short the Sasano
system. The system (1.1) is a two-degree-of-freedom non-autonomous Hamiltonian system with
the Hamiltonian
H = HV (x, y, t;α2 + α5, α1, α2 + 2α3 + α4) +HV (z, w, t;α5, α3, α4)
+
2yz((z − 1)w + α3)
t
, (1.2)
where HV (q, p, t; γ1, γ2, γ3) is the Hamiltonian associated with the fifth Painlevé equation, i.e.,
HV (q, p, t; γ1, γ2, γ3) =
q(q − 1)p(p+ t)− (γ1 + γ3)qp+ γ1p+ γ2tq
t
.
Hence the system (1.1) is considered as coupled Painlevé V systems in dimension 4. The non-
autonomous Hamiltonian system (1.1) can be turned into an autonomous one with three de-
grees of freedom by introducing two new dynamical variables: t and its conjugate variable −F .
The new Hamiltonian becomes
H̃ = H + F,
where H is given by (1.2). Then the extended Hamiltonian system (1.1) becomes
dx
ds
=
∂H̃
∂y
,
dy
ds
= −∂H̃
∂x
,
dz
ds
=
∂H̃
∂w
,
dw
ds
= −∂H̃
∂z
,
dt
ds
=
∂H̃
∂F
,
dF
ds
= −∂H̃
∂t
. (1.3)
The symplectic structure ω is canonical in the variables (x, y, z, w, t, F ), i.e., ω = dx∧dy+dz ∧
dw + dt ∧ dF .
In this paper, we are interested in the non-integrability of the Hamiltonian system (1.3).
Recall that from the theorem of Liouville–Arnold [1] this means the non-existence of three
first integrals f1 = H̃, f2, f3 functionally independent and in involution. We prove that the
Sasano system (1.1) is non-integrable by rational first integrals. Our approach comes under
the frame of the Morales–Ramis–Simó theory. This theory reduces the problem of integrabil-
ity of a given analytic Hamiltonian system to the problem of integrability of the variational
equation of this Hamiltonian system along one particular non-stationary solution. Since the
variational equations are linear ordinary differential equations their integrability is well defined
in the context of the differential Galois theory. The Morales–Ramis–Simó theory finds appli-
cations in the study of non-integrability of a huge range of dynamical systems like N -body
problems [4, 9, 16, 23, 35, 36], problems with homogeneous potentials [6, 10, 11], the Painlevé
equation and their q-analogues [7, 12, 18, 31, 32, 33], the higher-order analogues of Painlevé sys-
tems [34], as well as in the study of non-integrability of non-Hamiltonian systems [2, 5], in the
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 3
study of integrability in the Jacobi sense [15], in the study of the irreducibility of the Painlevé
equations [8], etc.
In this paper, we study the extended Sasano system (1.3) when α1 = α2 = α3 = 0, α4 = 1,
α0 = −α5. It turns out that the differential Galois group of the first variational equations is
a commutative group. To establish a non-integrable result, we find an obstruction to integra-
bility studying the linearized second normal variational equations (LNVE)2, which is a linear
homogeneous system of thirteen order. To determine the Galois group of such a higher-order lin-
ear system, we find a subsystem of the (LNVE)2, whose Galois group can be computed explicitly.
It turns out that this subsystem is a system with non-trivial Stokes phenomena at the infinity
point. Computing the corresponding Stokes matrices we deduce that the connected component
of the unit element of the differential Galois group of this subsystem and hence the Galois group
of the (LNVE)2 is not an Abelian group. Then the key result of this paper (Theorem 3.11 in
Section 3) states
Theorem 1.1. Assume that α1 = α2 = α3 = 0, α4 = 1, α0 = −α5, where α5 is arbitrary. Then
the Sasano system (1.1) is not integrable in the Liouville–Arnold sense by rational first integrals.
Using Bäcklund transformations of the Sasano system (1.1), which are rational canonical
transformations [28], we can extend the result of the key Theorem 1.1 to the main results of this
paper (Theorems 4.6 and 4.7 in Section 4).
Theorem 1.2. Let α be an arbitrary complex parameter, which is not an integer. Assume that
the parameters αj are either of the kind ±α + nj or of the kind lj , nj , lj ∈ Z in such a way
that 1− α0 − α1 and α4 + α5 are together of the kind ±α + mi, mi ∈ Z, i = 1, 2. Then the
Sasano system (1.1) is not integrable in the Liouville–Arnold sense by rational first integrals.
Theorem 1.3. Assume that all of the parameters αj, 0 ≤ j ≤ 5, are integer in such a way
that 1−α0−α1 and α4+α5 are together either even or odd integer. Then the Sasano system (1.1)
is not integrable in the Liouville–Arnold sense by rational first integrals.
In fact, the result of Theorem 1.2 contains the result of Theorem 1.3. We present Theorem 1.3
as an independent result because of the additional specification of the quantities 1 − α0 − α1
and α4 + α5 when α ∈ Z.
This paper is organized as follows. In the next section, we briefly review the basics of
the Morales–Rams–Simó theory of the non-integrability of the Hamiltonian systems and the
relation of the differential Galois theory to the linear systems of ordinary differential equations.
In Section 3, we prove non-integrability of the Sasano system (1.1) when α1 = α2 = α3 = 0,
α4 = 1 and α0 = −α5. In Section 4, using Bäcklund transformations of the Sasano system (1.1),
we extend the result of the Section 3 to the entire orbits of the parameters αj and establish the
main theorems of this paper.
2 Preliminaries
2.1 Non-integrability of Hamiltonian systems and differential Galois theory
In this subsection, we briefly recall Morales-Ruiz–Ramis–Simó theory of non-integrability of
Hamiltonian systems following [19, 20, 21, 22].
Let M be a symplectic analytical complex manifold of complex dimension 2n. Consider on M
a Hamiltonian system
ẋ = XH(x) (2.1)
4 Ts. Stoyanova
with a Hamiltonian H : M → C. Let x(t) be a particular solution of (2.1), which is not an
equilibrium point. Denote by Γ the phase curve corresponding to this solution. The first
variational equations (VE)1 of (2.1) along Γ are written
ξ̇ =
∂XH
∂x
(x(t))ξ, ξ ∈ TΓM. (2.2)
Using the Hamiltonian H, we can always reduce the degrees of freedom of the variational equa-
tions (2.2) by one in the following sense. Consider the normal bundle of Γ on the level variety
Mh = {x | H(x) = h}. The projection of the variational equations (2.2) on this bundle induces
the so called first normal variational equations (NVE)1 along Γ. The dimension of the (NVE)1
is 2n− 2. Assume now that x(t) is a rational non stationary particular solution of (2.1) and
let as above Γ be the phase curve corresponding to it. Assume also that the field K of the
coefficients of the (NVE)1 is the field of rational functions in t, that is K = C(t). Assume also
that t = ∞ is an irregular singularity for the (NVE)1. The entries of a fundamental matrix
solution of (NVE)1 define a Picard–Vessiot extension L1 of the field K. This in its turn defines
a differential Galois group G1 = Gal(L1/K). Then the main theorem of the Morales-Ruiz–Ramis
theory states [19, 20, 22].
Theorem 2.1 (Morales–Ramis). Assume that the Hamiltonian system (2.1) is completely in-
tegrable with rational first integrals in a neighbourhood of Γ, not necessarily independent on Γ
itself. Then the identity component (G1)
0 of the differential Galois group G1 = Gal(L1/K) is
Abelian.
The problem considered in this paper is one among many examples illustrating that the op-
posite is not true in general. That is if the connected component (G1)
0 of the unit element
of the differential Galois group Gal(L1/K) is Abelian, one cannot deduce that the correspond-
ing Hamiltonian system (2.1) is completely integrable. Beyond the first variational equations
Morales-Ruiz, Ramis and Simó suggest in [22] to use higher-order variational equations to solve
such integrability problems. Let as above x(t) be a particular rational non stationary solution
of the Hamiltonian system (2.1). We write the general solution as x(t, z), where z parametrizes
it near x(t) as x(t, z0) = x(t). Then we can write the system (2.1) as
ẋ(t, z) = XH(x(t, z)). (2.3)
Denote by x(k)(t, z), k ≥ 1 the derivatives of x(t, z) with respect to z and by X
(k)
H (x), k ≥ 1 the
derivatives of XH(x) with respect to x. By successive derivations of (2.3) with respect to z and
evaluations at z0, we obtain the so called k-th variational equations (VE)k along the solution x(t)
ẋ(k)(t) = X
(1)
H (x(t))x(k)(t) + P
(
x(1)(t), x(2)(t), . . . , x(k−1)(t)
)
. (2.4)
Here P denotes polynomial terms in the monomials of order |k| of the components of its ar-
guments. The coefficients of P depend on t through X
(j)
H (x(t)), j < k. For every k > 1,
the linear non-homogeneous system (2.4) can be arranged as a linear homogeneous system of
higher dimension by making the monomials of order |k| in P new variables and adding to (2.4)
their differential equations. If we restrict the system (2.4) to the variables that define the
(NVE)1, the corresponding linear homogeneous system is the so called k-th linearized nor-
mal variational equations (LNVE)k. The solutions of the chain of (LNVE)k define a chain
of Picard–Vessiot extensions of the main field K = C(t) of the coefficients of (NVE)1, i.e.,
we have K ⊂ L1 ⊂ L2 ⊂ · · · ⊂ Lk, where L1 is above, L2 is the Picard–Vessiot exten-
sion of K associated with (LNVE)2, etc. Then we can define the differential Galois groups
G1 = Gal(L1/K), G2 = Gal(L2/K), . . . , Gk = Gal(Lk/K). Assume as above that t = ∞ is an
irregular singularity for the (NVE)1 and therefore for (NVE)k for all k ≥ 2. Then the main
theorem of the Morales-Ruiz–Ramis–Simó theory states the following.
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 5
Theorem 2.2 (Morales–Ramis–Simó). If the Hamiltonian system (2.1) is completely integrable
with rational first integrals, then for every k ∈ N the connected component of the unit ele-
ment (Gk)
0 of the differential Galois group Gk = Gal(Lk/K) is Abelian.
From Theorem 2.2, it follows that if we find a group (Gk)
0, which is not Abelian, then the
Hamiltonian system (2.1) will be non-integrable by means of rational first integrals. Note that
this non-commutative group (Gk)
0 will be a solvable group. In this way, non-integrability in the
sense of the Hamiltonian dynamics will correspond to integrability in the Picard–Vessiot sense.
2.2 Differential Galois group of a linear system
of ordinary differential equations
In this subsection, we briefly recall some facts, notations and definitions from the differential
Galois theory, needed to compute the differential Galois group of a linear system with one
irregular and one regular singularity. We follow the works of van der Put, Mitschi, Singer and
Ramis [17, 24, 30, 37]. Throughout this paper, all angular directions and sectors are defined on
the Riemann surface of the natural logarithm.
Consider a linear system of ordinary differential equations of order n
υ̇ = A(t)υ, (2.5)
where A(t) ∈ GLn(C(t)).
Definition 2.3. The differential Galois group G of the system (2.5) over C(t) is the group of
all differential C(t)-automorphisms of a Picard–Vessiot extension of C(t) relative to (2.5). This
group is isomorphic to an algebraic subgroup of GLn(C) with respect to a fundamental matrix
solution of (2.5).
Assume that the system (2.5) has two singular points over CP1 taken at t = 0 and t = ∞.
Assume that the origin is a regular singularity while t = ∞ is a non-resonant irregular singularity
of Poincaré rank 1. Denote by S the set of singular points of the system (2.5), that is, S = {0,∞}.
If we replace in Definition 2.3 the field C(t) with the field of germs of meromorphic functions
at a ∈ S, we define the so called local differential Galois group Ga of (2.5). In what follows, we
present effective theorems for computing the local differential Galois groups Ga, a ∈ S of the
system (2.5).
Let Φ(t) be a local fundamental matrix solution near the origin of the system (2.5). The fol-
lowing result of Schlesinger [29] describes the local differential Galois group at the origin of the
system (2.5).
Theorem 2.4 (Schlesinger). Under the above assumptions the monodromy group around the
origin with respect to the fundamental matrix solution Φ(t) is a Zariski dense subgroup of the
differential Galois group G of the system (2.5).
Since we prefer to work with an irregular singularity at the origin to at t = ∞, we make the
change t = 1/τ in the system (2.5). This transformation takes the system (2.5) into the system
υ′ = A(τ)υ, ′ =
d
dτ
, (2.6)
for which the origin is a non-resonant irregular singularity of Poincaré rank 1. Denote by C(τ),
C((τ)) and C{τ} the differential fields of rational functions, formal power series and convergent
power series, respectively. Note that
C(τ) ⊂ C{τ} ⊂ C((τ)).
6 Ts. Stoyanova
In what follows, we determine the local differential Galois groups of the system (2.6) around
the origin.
From the Hukuhara–Turrittin theorem [38], it follows that the system (2.6) admits a formal
fundamental matrix solution at the origin of the form
Ψ̂(τ) = Ĥ(τ)τΛ exp
(
Q
τ
)
, (2.7)
where
Λ = diag(λ1, λ2, . . . , λn), Q = diag(q1, q2, . . . , qn), Ĥ(t) ∈ GLn(C((t)))
with λj , qj ∈ C, j = 1, . . . , n. Consider the system (2.6) and its formal fundamental matrix
solution Ψ̂(τ) over the field C((τ)).
Definition 2.5. With respect to the formal fundamental matrix solution Ψ̂(τ) from (2.7), we
define the formal monodromy matrix M̂0 ∈ GLn(C) around the origin as
Ψ̂
(
τ.e2πi
)
= Ψ̂(τ)M̂0.
In particular,
M̂0 = e2πiΛ.
Definition 2.6. With respect to the formal fundamental matrix solution Ψ̂(τ) from (2.7) we
define the exponential torus T as the differential Galois group Gal(E/F ), where
F = C((τ))
(
τλ1 , τλ2 , . . . , τλn
)
and E = F
(
eq1/τ , eq2/τ , . . . , eqn/τ
)
.
We may consider T as a subgroup of (C∗)n. The Zariski closure of the group generated by the
formal monodromy matrix and exponential torus yields the so called formal differential Galois
group at the origin of the system (2.6) (see [30, 37]).
Consider now the system (2.6) over the field C{τ}. In general, the entries ĥij(τ), 1 ≤ i, j ≤ n
of the matrix Ĥ(τ) in (2.7) are either divergent or convergent power series in τ . The existence of
divergent power series entries in Ĥ(τ) ensures an observation of a non-trivial Stokes phenomenon
at the origin.
Definition 2.7. Under the above notations for every divergent power series ĥij(τ), we define
a set Θj of admissible singular directions θji, 0 ≤ θji < 2π, where θji is the bisector of the
maximal angular sector
{
Re
( qi−qj
τ
)
< 0
}
. In particular,
Θj = {θji, 0 ≤ θji < 2π, θji = arg(qj − qi), 1 ≤ i, j ≤ n, i ̸= j}.
In order to compute the analytic invariants at the origin of the system (2.6), we have to
lift the formal fundamental matrix solution Ψ̂(τ) from (2.7) to such an actual. To solve this
problem, in this paper we utilize the summability theory. The application of the summability
theory to ordinary differential equations generalizes the theorem of Hukuhara–Turrittin to the
following theorem of Ramis [25].
Theorem 2.8. In the formal fundamental matrix solution at the origin Ψ̂(τ) from (2.7) the
entries of the matrix Ĥ(τ) are 1-summable along any non-singular direction θ. If we denote
by Hθ(τ) the 1-sum of the matrix Ĥ(τ) along θ, then Ψθ(τ) = Hθ(τ)τ
Λ exp(Q/τ) is an actual
fundamental matrix solution at the origin of the system (2.6).
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 7
Since this paper is not devoted to the summability theory, rather we only use it, we will not
consider it in details. For the needed facts, notation and definitions, we refer to the works of
Loday-Richaud [14], as well as the works of Ramis [24, 25].
Let ε > 0 be a small number. Let θ− ε and θ+ ε be two non-singular neighboring directions
to the singular direction θ ∈ Θj . Let Ψθ−ε(τ) and Ψθ+ε(τ) be the actual fundamental matrix
solutions at the origin of the system (2.6) corresponding to the directions θ− ε and θ+ ε in the
sense of Theorem 2.8.
Definition 2.9. With respect to the actual fundamental matrix solutions Ψθ−ε(τ) and Ψθ+ε(τ)
the Stokes matrix Stθ ∈ GLn(C) related to the singular direction θ is defined as
Stθ = (Ψθ+ε(τ))
−1Ψθ−ε(τ).
The next theorem of Ramis [24] determines the differential Galois group at the origin of the
system (2.6) over the field C(τ}.
Theorem 2.10 (Ramis). The differential Galois group at the origin of the system (2.6) over
C{τ} is the Zariski closure of the group generated by the formal differential Galois group at the
origin and the collection of the Stokes matrices {Stθ} for all singular directions θ.
For more details about the relation between the Stokes phenomenon and the differential
Galois theory, we refer to the very recent work of Ramis [26]. We make note that one can
introduce the differential Galois group at t = ∞ of the system (2.5) in the same way.
Let t0 be a base point of CP1\S and let Σt0 denote an analytic germ of a fundamental matrix
solution of (2.5) at t0. Let Ua, a ∈ S, be an open disc with center a, together with a local
parameter ta at a, and such that Ua∩S = {a}. Let da be a fixed ray from a in Ua, together with
a point ba ∈ da in Ua and a path γa from t0 to ba. Analytic continuation of Σt0 along γa and da
provides an analytic germ Σa of fundamental matrix solution on a germ of open sector with
vertex a, bisected by da. Let Ga be the local differential Galois group of the system (2.5) over
the field of germs of meromorphic function at a with respect to Σa. If we conjugate elements
of Ga by the analytic continuation described above, we get an injective morphism of algebraic
groups Ga ↪→ G with respect to the representation of these groups in GLn(C) given by Σa
and Σt0 , respectively. In this way all Ga, a ∈ S, can be simultaneously identified with closed
subgroups of G. Then we have the following important result of Mitschi [17, Proposition 1.3].
Theorem 2.11 (Mitschi). The differential Galois group G of the system (2.5) is topologically
generated in GLn(C) by the local differential Galois groups Ga, where a runs over S.
3 Non-integrability for α1 = α2 = α3 = 0, α4 = 1, α0 = −α5
In this section, we deal with the non-integrability of the Hamiltonian system (1.3) when α1 =
α2 = α3 = 0, α4 = 1, α0 = −α5. Denote α = α5. For these values of the parameters the
autonomous Hamiltonian system (1.3) becomes
dx
ds
=
2x2y
t
+ x2 − 2xy
t
−
(
1 +
1 + α
t
)
x+
α
t
+
2z(z − 1)w
t
,
dy
ds
= −2xy2
t
+
y2
t
− 2xy +
(
1 +
1 + α
t
)
y,
dz
ds
=
2z2w
t
+ z2 − 2zw
t
−
(
1 +
1 + α
t
)
z +
α
t
+
2yz(z − 1)
t
,
dw
ds
= −2zw2
t
+
w2
t
− 2zw +
(
1 +
1 + α
t
)
w − 2y(−w + 2zw)
t
,
8 Ts. Stoyanova
dt
ds
= 1,
dF
ds
=
1
t2
(x(x− 1)y(y + t) + z(z − 1)w(w + t)− (1 + α)(xy + zw) + α(y + w))
− x(x− 1)y
t
− z(z − 1)w
t
.
We choose
x = z =
α
s
, y = w = 0, t = s, F = 0 (3.1)
as a non-equilibrium particular solution, along which we will write the variational equations.
Because of the equation dt
ds = 1, from here on we use t instead of s.
For the first normal variational equations (NVE)1 along the solution (3.1), we obtain the
system
ẋ1 =
(
−1 +
α− 1
t
)
x1 +
(
2α2
t3
− 2α
t2
)
y1 +
(
2α2
t3
− 2α
t2
)
w1,
ż1 =
(
−1 +
α− 1
t
)
z1 +
(
2α2
t3
− 2α
t2
)
w1 +
(
2α2
t3
− 2α
t2
)
y1,
ẇ1 =
(
1− α− 1
t
)
w1,
ẏ1 =
(
1− α− 1
t
)
y1. (3.2)
Note that the (NVE)k, k ∈ N of the system (1.3) along the solution (3.1) are nothing but the
(VE)k, k ∈ N, of the system (1.1) along the solution x = z = α
t , y = w = 0 for α1 = α2 = α3 = 0,
α4 = 1, α0 = −α5 = −α.
The system (3.2) is solvable in quadratures and therefore its differential Galois group G1 is
a solvable subgroup in GL4(C).
Theorem 3.1. The connected component (G1)
0 of the unit element of the differential Galois
group G1 of the (NVE)1 is Abelian.
Proof. The (NVE)1 have two singular points over CP1: the points t = 0 and t = ∞. The ori-
gin is a regular singularity while t = ∞ is an irregular singularity. The system (3.2) admits
a fundamental matrix solution Φ(t) in the form
Φ(t) =
e−ttα−1 0 −αett−α−1 −αett−α−1
0 e−ttα−1 −αett−α−1 −αett−α−1
0 0 ett−α+1 0
0 0 0 ett−α+1
.
We will compute the Galois group G1 of the (NVE)1 with respect to this fundamental matrix
solution. In this case, from Theorem 2.11, it follows that the differential Galois group G1 of the
system (3.2) is generated topologically by the local Galois groups G0 and G∞, corresponding to
the singularities t = 0 and t = ∞, respectively.
Let us first determine the group G0. In a neighborhood of the origin, the above solution Φ(t)
is written as
Φ(t) = P (t)tA,
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 9
where P (t) is the holomorphic matrix
P (t) =
e−t 0 −αet −αet
0 e−t −αet −αet
0 0 t2et 0
0 0 0 t2et
.
For the constant matrix A we have that A = diag(α−1, α−1,−α−1,−α−1). From Theorem 2.4,
it follows that the Galois group G0 over C(t) is generated topologically by the monodromy
matrix M0 around the origin. With respect to the fundamental matrix solution Φ(t), we obtain
M0 = e2πiA =
e2πiα 0 0 0
0 e2πiα 0
0 0 e−2πiα 0
0 0 0 e−2πiα
.
When α ∈ Q but α /∈ Z, the group generated by M0 is not connected but it is a finite and cyclic
group. In this case,
G0 =
ν−1 0 0 0
0 ν−1 0 0
0 0 ν 0
0 0 0 ν
, ν is a root of unity
, (G0)
0 = {I4},
where I4 is the identity matrix. When α ∈ Z, we have that G0 = (G0)
0 = {I4}. When α /∈ Q,
the group generated by M0 is a connected group and
G0 = (G0)
0 =
ν−1 0 0 0
0 ν−1 0 0
0 0 ν 0
0 0 0 ν
, ν ∈ C
.
Consider now the (NVE)1 and the fundamental matrix solution Φ(t) at t = ∞. In a neigh-
borhood of the irregular singularity t = ∞, the matrix Φ(t) is written as
Φ(t) = H1(t)t
Λ1 exp(Q1t),
where H1(t) is the holomorphic matrix
H1(t) =
1 0 −αt−2 −αt−2
0 1 −αt−2 −αt−2
0 0 1 0
0 0 0 1
.
The matrices Λ1 and Q1 are given by
Λ1 = diag(α− 1, α− 1,−α,−α), Q1 = diag(−1,−1, 1, 1).
Since we do not observe non-trivial Stokes phenomena, the local Galois group G∞ is generated
topologically by the formal monodromy M̂∞ and the exponential torus T∞. The formal mon-
odromy corresponding to a loop around t = ∞ is nothing but (M0)
−1, that is, M̂∞ = (M0)
−1.
Therefore, we can consider the local differential Galois group G0 at the origin as a subgroup
of the local differential Galois group G∞ at t = ∞. Thus the Galois group G1 of the (NVE)1
10 Ts. Stoyanova
coincides with the local differential Galois group G∞ at t = ∞. For the exponential torus, we
have
T∞ =
λ−1 0 0 0
0 λ−1 0 0
0 0 λ 0
0 0 0 λ
,
where λ ∈ C∗.
As a result, we find that when α ∈ Q the connected component (G1)
0 of the differential
Galois group G1 coincides with T∞. When α /∈ Q the group (G1)
0 is generated by M̂∞ and T∞.
Summarily, the group (G1)
0 is defined as
(G1)
0 =
µ−1 0 0 0
0 µ−1 0 0
0 0 µ 0
0 0 0 µ
, µ ∈ C∗
,
which is an Abelian group. ■
For the second normal variational equations (NVE)2 along the solution (3.1), we obtain the
non-homogeneous system
ẋ2 =
(
−1 +
α− 1
t
)
x2 +
(
2α2
t3
− 2α
t2
)
y2 +
(
2α2
t3
− 2α
t2
)
w2 + x21
+
(
4α
t2
− 2
t
)
x1y1 +
(
4α
t2
− 2
t
)
w1z1,
ẏ2 =
(
1− α− 1
t
)
y2 − 2x1y1 +
(
1
t
− 2α
t2
)
y21,
ż2 =
(
−1 +
α− 1
t
)
z2 +
(
2α2
t3
− 2α
t2
)
w2 +
(
2α2
t3
− 2α
t2
)
y2 + z21
+
(
4α
t2
− 2
t
)
y1z1 +
(
4α
t2
− 2
t
)
w1z1,
ẇ2 =
(
1− α− 1
t
)
w2 −
(
4α
t2
− 2
t
)
y1w1 − 2w1z1 +
(
1
t
− 2α
t2
)
w2
1.
Introducing more 9 new variables x21, x1y1, x1w1, w1z1, w
2
1, w1y1, y
2
1, z
2
1 , z1y1 and their dif-
ferential equations, we extend the (NVE)2 to the (LNVE)2 [22]. The (LNVE)2 is a system of
thirteenth order. The very high order of the (LNVE)2 make the problem of the description
of its differential Galois group too complicated. Fortunately, it is not necessary to study the
whole (LNVE)2. If we find a subsystem of (LNVE)2, for which the connected component G0 of
the unit element of the corresponding differential Galois group is not Abelian and so is (G2)
0.
For this reason, from here on we study the differential Galois group of the following fourth-
order linear homogeneous system:
ẇ2 =
(
1− α− 1
t
)
w2 − 2p−
(
4α
t2
− 2
t
)
q +
(
1
t
− 2α
t2
)
v,
ṗ =
(
2α2
t3
− 2α
t2
)
q +
(
2α2
t3
− 2α
t2
)
v,
q̇ = 2
(
1− α− 1
t
)
q,
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 11
v̇ = 2
(
1− α− 1
t
)
v, (3.3)
where we have denoted p := w1z1, q := y1w1, v := w2
1. The system (3.3) as the (NVE)1 has two
singular points over CP1: t = 0 and t = ∞. The origin is a regular singularity, while t = ∞ is
an irregular singularity of Poincaré rank 1.
In order to determine the local differential Galois group at t = ∞ of the system (3.3), we
make the change t = 1/τ . This transformation takes the system (3.3) into the system
w′
2 =
(
α− 1
τ
− 1
τ2
)
w2 +
2
τ2
p+
(
4α− 2
τ
)
q +
(
2α− 1
τ
)
v,
p′ =
(
2α− 2α2τ
)
q +
(
2α− 2α2τ
)
v,
q′ = 2
(
α− 1
τ
− 1
τ2
)
q,
v′ = 2
(
α− 1
τ
− 1
τ2
)
v, (3.4)
where ′ = d
dτ . Now the origin is an irregular singularity of Poincaré rank 1 for the system (3.4).
We note that from here on we use the standard notations (a)n and (a)(n) for the falling and the
rising factorials
(a)n = a(a− 1)(a− 2) · · · (a− n+ 1), (a)0 = 1,
(a)(n) = a(a+ 1)(a+ 2) · · · (a+ n− 1), a(0) = 1,
respectively.
Proposition 3.2. The system (3.4) possesses an unique formal fundamental matrix solution at
the origin in the form
Ψ̂(τ) = Ĥ(τ)τΛ exp
(
Q
τ
)
,
where the matrices Λ and Q are given by
Λ = diag(α− 1, 0, 2α− 2, 2α− 2), Q = diag(1, 0, 2, 2).
The matrix Ĥ(τ) is defined as
Ĥ(τ) =
1 2 + φ̂(τ) 2τ ϕ̂(τ)
0 1 −ατ2 −ατ2
0 0 1 0
0 0 0 1
.
The elements φ̂(τ) and ϕ̂(τ) are defined as follows:
1. If α ∈ N, the function φ̂(τ) is the polynomial
φ̂(τ) = 2(α− 1)τ + 2(α− 1)(α− 2)τ2 + · · ·+ 2(α− 1)!τα−1. (3.5)
Otherwise, φ̂(τ) is given by the following divergent power series:
φ̂(τ) = 2
∞∑
n=1
(α− 1)nτ
n.
12 Ts. Stoyanova
2. If α ∈ Z≤0, the function ϕ̂(τ) is the polynomial
ϕ̂(τ) = τ + ατ2 + α(α+ 1)τ3 + · · ·+ (−1)−α(−α)!τ−α+1. (3.6)
Otherwise, ϕ̂(τ) is given by the following divergent power series:
ϕ̂(τ) =
∞∑
n=0
α(n)τn+1.
Proof. The formulas
p(τ) = C3 − C2αe
2
τ τ2α − C1αe
2
τ τ2α, q(τ) = C2e
2
τ τ2(α−1), v(τ) = C1e
2
τ τ2(α−1),
where C1, C2, C3 are constant of integration, give the general solutions of the last three equations
of the system (3.4). To build a local fundamental matrix solution Ψ̂(τ) at the origin, we use that
each column of such a matrix is a solution of the system (3.4). Denote by Ψ̂j(τ), j = 1, 2, 3, 4,
the columns of the matrix Ψ̂(τ). Then
Ψ̂1(τ) =
ŵ
(1)
2 (τ)
0
0
0
,
where ŵ
(1)
2 (τ) is a solution of the equation
w′
2 =
(
α− 1
τ
− 1
τ2
)
w2.
We choose ŵ
(1)
2 (τ) = e
1
τ τα−1. For the second column Ψ̂2(τ), we have
Ψ̂2(τ) =
ŵ
(2)
2 (τ)
1
0
0
,
where ŵ
(2)
2 (τ) is a formal solution of the equation
w′
2 =
(
α− 1
τ
− 1
τ2
)
w2 +
2
τ2
. (3.7)
The equation (3.7) admits a formal solution near the origin in the form
ŵ
(2)
2 (τ) = 2 + 2
∞∑
n=1
(α− 1)nτ
n,
which is a polynomial when α ∈ N. When α /∈ N, the solution ŵ
(2)
2 (τ) is a divergent power
series.
For the next column, we have
Ψ̂3(τ) =
ŵ
(3)
2 (τ)
−αe
2
τ τ2α
e
2
τ τ2(α−1)
0
,
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 13
where ŵ
(3)
2 (τ) is a solution of the equation
w′
2 =
(
α− 1
τ
− 1
τ2
)
w2 + 2αe
2
τ τ2α−2 − 2e
2
τ τ2α−3.
We choose ŵ
(3)
2 (τ) = 2e
2
τ τ2α−1 .
For the last column Ψ̂4(τ), we have
Ψ̂4(τ) =
ŵ
(4)
2 (τ)
−αe
2
α τ2α
0
e
2
τ τ2(α−1)
,
where ŵ
(4)
2 (τ) is a solution of the equation
w′
2 =
(
α− 1
τ
− 1
τ2
)
w2 − e
2
τ τ2α−3. (3.8)
Looking for a solution of the equation (3.8) in the form ŵ
(4)
2 (τ) = e
2
τ τ2α−3ĝ(τ), we find that ĝ(τ)
must satisfy the equation
τ2ĝ′ = (1− (α− 2))ĝ − τ2. (3.9)
The equation (3.9) admits a formal solution near the origin in the form
ĝ(τ) = τ
∞∑
n=0
α(n)τn+1,
which is a polynomial when α ∈ Z≤0. Otherwise, ĝ(τ) is a divergent power series.
Fitting together the so building columns Ψ̂j(τ), j = 1, 2, 3, 4, we complete the proof. ■
Now we can determine explicitly the formal invariants at the origin of the system (3.4).
Proposition 3.3. With respect to the formal fundamental matrix solution Ψ̂(τ) introduced by
Proposition 3.2, the exponential torus T and the formal monodromy M̂0 at the origin of the
system (3.4) are given by
T =
λ 0 0 0
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗
, M̂0 = e2πiΛ =
e2πiα 0 0 0
0 1 0 0
0 0 e4πiα 0
0 0 0 e4πiα
.
The application of Definition 2.7 to the divergent power series φ̂(τ) and ϕ̂(τ) gives the fol-
lowing sets of admissible singular directions:
Θ2 = {θ = arg(0− 1) = arg(0− 2) = π}
for the series φ̂(τ) and
Θ3 = {θ = arg(2− 1) = arg(2− 0) = 0}
for the series ϕ̂(τ).
In the next lemma, we compute the 1-sums of the divergent power series φ̂(τ) and ϕ̂(τ). These
1-sums illustrate explicitly the dependence of the power series φ̂(τ) and ϕ̂(τ) on the suggested
admissible singular direction. Denote a = Re(α) and by [|a|], [a] the integer parts of the real
numbers |a| and a, respectively.
14 Ts. Stoyanova
Lemma 3.4. Under the above notations, we have
1. Assume that α /∈ N. Then for every direction θ ̸= π the function
φθ(τ) = 2(α− 1)
∫ +∞eiθ
0
(1 + ζ)α−2e−
ζ
τ dζ
defines the 1-sum of the power series φ̂(τ) in such a direction.
2. Assume that α /∈ Z≤0. Then for every direction θ ̸= 0 the function
ϕθ(τ) =
∫ +∞eiθ
0
(1− ζ)−αe−
ζ
τ dζ
defines the 1-sum of the power series ϕ̂(τ) in such a direction.
When Re(α) ≤ 2 (resp. Re(α) ≥ 0) the function φθ(τ) (resp. ϕθ(θ)) is a holomorphic function
in the open unlimited disc
Dθ =
{
τ ∈ C | Re
(
eiθ
τ
)
> 0
}
for any direction θ ̸= π (resp. θ ̸= 0). Otherwise, they are holomorphic functions in the open
bounded disc
Dθ(1) =
{
τ ∈ C | Re
(
eiθ
τ
)
> 1
}
.
Proof. Consider the divergent power series φ̂(τ) and ϕ̂(τ) defined by Proposition 3.2. For the
corresponding formal Borel transforms, we obtain the power series
B̂1φ̂(ζ) = 2
∞∑
n=0
(α− 1)n+1
ζn
n!
= 2(α− 1)
∞∑
n=0
(α− 2)n
ζn
n!
, B̂1ϕ̂(ζ) =
∞∑
n=0
α(n) ζ
n
n!
.
Both of the series B̂1φ̂(ζ) and B̂1ϕ̂(ζ) are convergent near the origin of the Borel ζ-plane with
finite radiuses of convergence. Therefore, both of the divergent series φ̂(τ) and ϕ̂(τ) are Gevrey
series of order 1. The functions
φ(ζ) = 2(α− 1)(1 + ζ)α−2, ϕ(ζ) = (1− ζ)−α
present the sums of the power series B̂1φ̂(ζ) and B̂1ϕ̂(ζ), respectively.
Consider the function (1 + ζ)α−2. We have that
∣∣(1 + ζ)α−2
∣∣ = A|1 + ζ|Re(α)−2, where
A = e−Im(α) arg(1+ζ). Let θ = arg(ζ). If Re(α)− 2 ≤ 0, then
A
|1 + ζ|2−Re(α)
≤ A
when cos θ ≥ 0, while
A
|1 + ζ|2−Re(α)
≤ A
| sin θ|2−Re(α)
when cosα < 0. If Re(α)− 2 > 0, then
A|1 + ζ|Re(α)−2 ≤ Be|ζ|
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 15
for an appropriate constant B > 0. Therefore, the function φ(ζ) is of exponential size at most 1
at ∞ along any direction θ ̸= π from 0 to +∞eiθ. Moreover, the function φ(ζ) is continued
analytically along any such a direction. Then the corresponding Laplace transform (Lθφ)(τ) is
well defined and gives the 1-sum of the divergent power series φ̂(τ) in such a direction. If we
denote by φθ(τ) this 1-sum, we have that
φθ(τ) = 2(α− 1)
∫ +∞eiθ
0
(1 + ζ)α−2e−
ζ
τ dζ
for every θ ̸= π. From the above estimates, it follows that when Re(α)− 2 ≤ 0, the 1-sum φθ(τ)
is a holomorphic function in the open unlimited disc
Dθ =
{
τ ∈ C | Re
(
eiθ
τ
)
> 0
}
, (3.10)
whose opening is π. When Re(α) − 2 > 0, the 1-sum is a holomorphic function in the open
bounded disc
Dθ(1) =
{
τ ∈ C | Re
(
eiθ
τ
)
> 1
}
(3.11)
with opening < π.
Using similar arguments, we find that for any direction θ ̸= 0 the Laplace transform
ϕθ(τ) =
∫ +∞eiθ
0
(1− ζ)−αe−
ζ
τ dζ
defines the 1-sum of the divergent power series ϕ̂(τ) in such a direction. When Re(α) ≥ 0,
the 1-sum ϕθ(τ) is a holomorphic function in the disc Dθ from (3.10). Otherwise, the 1-sum is
a holomorphic function in the disc Dθ(1) from (3.11). ■
Remark 3.5. Let I = (−π, π) ⊂ R and J = (0, 2π) ⊂ R. When we move the direction θ ∈ I, the
holomorphic functions φθ(τ) glue together analytically and define a holomorphic function φ̃(τ)
on a sector D̃1 with opening 3π, −3π
2 < arg(τ) < 3π
2 when Re(α) ≤ 2 or on a sector
D̃1(1) =
⋃
θ∈I
Dθ(1)
with opening> π when Re(α) > 2. Similarly, when we move the direction θ ∈ J , the holomorphic
functions ϕθ(τ) glue together analytically and define a holomorphic function ϕ̃(τ) on a sector D̃2
with opening 3π, −π
2 < arg(τ) < 5π
2 when Re(α) ≥ 0 or on a sector
D̃2(1) =
⋃
θ∈J
Dθ(1)
with opening > π when Re(α) < 0. On these sectors, the functions φ̃(τ) and ϕ̃(τ) are asymptotic
to the power series φ̂(τ) and ϕ̂(τ), respectively, in Gevrey 1-sense and define the 1-sums of these
power series there. Their restrictions on C∗ are multivalued functions. In any direction θ ̸= π
(resp. θ ̸= 0) the function φ̃(τ)
(
resp. ϕ̃(τ)
)
has only one value which coincides with the func-
tion φθ(τ) (resp. ϕθ(τ)) defined by Lemma 3.4. Near the singular direction θ = π (resp. θ = 0) the
function φ̃(τ)
(
resp. ϕ̃(τ)
)
has two different values: φ+
π (τ) = φπ+ε(τ)
(
resp. ϕ+
0 (τ) = ϕ0+ε(τ)
)
and φ−
π (τ) = φπ−ε(τ) (resp. ϕ
−
0 (τ) = ϕ0−ε(τ)) for a small number ε > 0.
16 Ts. Stoyanova
Replacing the divergent power series entries of the matrix Ĥ(τ) with their 1-sums, we get
an actual fundamental matrix solution at the origin of the system (3.4). More precisely, denote
F (τ) = τΛ exp
(Q
τ
)
.
Proposition 3.6. For every non-singular direction θ, the system (3.4) possesses an unique
actual fundamental matrix solution at the origin in the form
Ψθ(τ) = Hθ(τ)Fθ(τ), (3.12)
where Fθ(τ) is the branch of the matrix F (τ) for θ = arg(τ). The matrix Hθ(τ) is given by
Hθ(τ) =
1 2 + h12(τ) 2τ h14(τ)
0 1 −ατ2 −ατ2
0 0 1 0
0 0 0 1
.
The entries h12(τ) and h14(τ) of the matrix Hθ(τ) are defined as follows:
1. If α ∈ N, then h14(τ) = ϕθ(τ), where ϕθ(τ) is defined by Lemma 3.4 and extended by
Remark 3.5. The element h12(τ) coincides with the function φ̂(τ) from (3.5).
2. If α ∈ Z≤0, then h12(τ) = φθ(τ), where φθ(τ) is defined by Lemma 3.4 and extended by
Remark 3.5. The element h14(τ) coincides with the function ϕ̂(τ) from (3.6).
3. If α /∈ Z, then h12(τ) = φθ(τ), h14(τ) = ϕθ(τ), where φθ(τ) and ϕθ(τ) are defined by
Lemma 3.4 and extended by Remark 3.5.
Near the singular direction θ = 0 or θ = π, the system (3.4) possesses two different actual
fundamental matrix solutions at the origin in the form
Ψ+
0 (τ) = Ψ0+ε(τ) and Ψ−
0 (τ) = Ψ0−ε(τ)
or
Ψ+
π (τ) = Ψπ+ε(τ) and Ψ−
π (τ) = Ψπ−ε(τ),
where Ψ0±ε(τ) and Ψπ±ε(τ) are defined by (3.12) for a small number ε > 0.
Now we can compute the analytic invariants at the origin of the system (3.4).
Theorem 3.7. With respect to the actual fundamental matrix solution at the origin, defined by
Proposition 3.6, the system (3.4) has a Stokes matrix Stπ in the form
Stπ =
1 µ1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
.
The multiplier µ1 is defined as follows:
1. If Re(α− 1) > 0, then
µ1 = 4i(α− 1)(−1)α−1 sin((2− α)π)Γ(α− 1).
2. If Re(α− 1) ≤ 0 but α /∈ Z≤0, then
µ1 =
4πi(α− 1)(−1)α−1
Γ(2− α)
.
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 17
3. If α ∈ Z≤0, then
µ1 =
4πi(−1)−α
(−α)!
.
Similarly, with respect to the actual fundamental matrix solution at the origin, defined by
Proposition 3.6, the system (3.4) has a Stokes matrix St0 in the form
St0 =
1 0 0 µ2
0 1 0 0
0 0 1 0
0 0 0 1
.
The multiplier µ2 is defined as follows:
1. If Re(1− α) > 0, then
µ2 = −2i sin((1− α)π)Γ(1− α).
2. If Re(1− α) ≤ 0 but α /∈ N, then
µ2 = − 2πi
Γ(α)
.
3. If α ∈ N, then
µ2 = − 2πi
(α− 1)!
.
Proof. From the Definition 2.9, it follows that the multiplier µ1 is computed by comparing the
solutions φ−
π (τ) and φ+
π (τ). Denote
J1 = φ−
π (τ)− φ+
π (τ).
Then
J1 = 2(α− 1)
∫
γ
(1 + ζ)α−2e−ζ/τdζ for
π
2
< arg(τ) <
3π
2
,
where γ = (π − ε) − (π + ε) for a small number ε > 0. Assume that α /∈ Z≤0. Then without
changing the integral, we can deform the path γ into a Hankel type contour γ1 winding around
the branch cut on R− of the function (1 + ζ)α−2, starting on −∞, encircling −1 in the positive
sense and returning to −∞. Then J1 becomes
J1 = 2(α− 1)
∫
γ1
(1 + ζ)α−2e−ζ/τdζ.
The transformation 1 + ζ = u takes the contour γ1 into a Hankel type contour γ2 going along
the branch cut on R− of the function uα−2, starting on −∞, encircling 0 in the positive sense
and backing to −∞. Then we have
J1 = 2(α− 1)e1/τ
∫
γ2
uα−2e−u/τdu.
18 Ts. Stoyanova
Now the change u/τ = −η takes the contour γ2 into itself and we find that
J1 = 2(α− 1)(−1)α−1τα−1e1/τ
∫
γ2
ηα−2eηdη.
To obtain the formula for µ1, we use the well-known Hankel’s representation of the Gamma
function Γ(1− α) and reciprocal Gamma function 1/Γ(α) when α ̸= 0,−1,−2, . . . as a contour
integral (see [3, 1.6 (1) and 1.6 (2)])∫
γ2
η−αeηdη = 2i sin(απ)Γ(1− α),
1
2πi
∫
γ2
η−αeηdη =
1
Γ(α)
.
Hence
φ−
π (τ)− φ+
π (τ) = 4i(α− 1)(−1)α−1 sin((2− α)π)Γ(α− 1)τα−1e1/τ
when Re(α− 1) > 0, and
φ−
π (τ)− φ+
π (τ) =
4πi(α− 1)(−1)α−1
Γ(2− α)
τα−1e1/τ
when Re(α− 1) ≤ 0 but α /∈ Z≤0.
Assume now that α ∈ Z≤0. Then from the Cauchy’s differential formula it follows that
φ−
π (τ)− φ+
π (τ) =
4(α− 1)πi
(1− α)!
[
D1−α
ζ
(
e−
ζ
τ
)
|ζ=−1
]
=
4πi(−1)−α
(−α)!
τα−1e
1
τ ,
where Dn
ζ = dn
dζn .
In a similar manner comparing the solutions τ2α−2e2/τϕ−
0 (τ) and τ2α−2e2/τϕ+
0 (τ), one can
derive the multiplier µ2. ■
Now we can describe the local differential Galois group G at the origin of the system (3.4).
Theorem 3.8. With respect to the formal and actual fundamental matrix solutions, given by
Propositions 3.2 and 3.6, the connected component G0 of the unit element of the local differential
Galois group G at the origin of the system (3.4) is defined as follows:
1. If α ∈ N, then
G0 =
λ 0 0 µ
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ ∈ C
.
2. If α ∈ Z≤0, then
G0 =
λ µ 0 0
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ ∈ C
.
3. If α /∈ Z, then
G0 =
λ µ 0 ν
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ, ν ∈ C
.
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 19
Proof. From Theorem 2.10, it follows that the local differential Galois group G at τ = 0 of
the system (3.4) is the Zariski closure of the group generated by the formal differential Galois
group and the Stokes matrices Stπ and St0. Since when α ∈ Z the formal monodromy M̂0
is equal to the identity matrix I4 then the formal differential Galois group coincides with the
exponential torus T . Therefore, in this case G is generated by the exponential torus and the
Stokes matrices Stπ and St0. Since when α ∈ N the Stokes matrix Stπ is equal to I4 the Galois
group G is generated by the exponential torus T and the Stokes matrix St0. Hence for α ∈ N
the differential Galois group coincides with its connected component G0 of the unit element and
G = G0 =
λ 0 0 µ
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ ∈ C
.
Similarly, when α ∈ Z≤0 we have that G is generated topologically by T and Stπ and
G = G0 =
λ µ 0 0
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ ∈ C
.
When α ∈ Q but α /∈ Z, the formal differential Galois group is not connected since the group
generated by M̂0 is not connected. However, in this case the connected component of the unit
element of the formal differential Galois group coincides with T . Therefore, in this case G does
not coincides with G0 but G0 is generated by T and the Stokes matrices Stπ and St0. Hence
G0 =
λ µ 0 ν
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ, ν ∈ C
.
In the last case when α /∈ Q, we have that
G = G0 =
λ µ 0 ν
0 1 0 0
0 0 λ2 0
0 0 0 λ2
, λ ∈ C∗, µ, ν ∈ C
.
This ends the proof. ■
Directly from Theorem 3.8, we obtain the following important result
Theorem 3.9. The connected component of the unit element of the differential Galois group of
the system (3.3) is not Abelian.
Proof. If we prove that the connected component G0 of the unit element of the local differential
Galois group at the origin of the system (3.4) is not Abelian group, then we will have that the
connected component of the unit element of the differential Galois group of the system (3.3) will
be not Abelian too. To prove that G0, defined by Theorem 3.8, is not an Abelian group it is
enough to show that the matrices
Tλ =
λ 0 0 0
0 1 0 0
0 0 λ2 0
0 0 0 λ2
and Sµ,ν =
1 µ 0 ν
0 1 0 0
0 0 1 0
0 0 0 1
do not commute.
20 Ts. Stoyanova
When µ and ν do not vanish together, the commutator between Sµ,ν and Tλ
Sµ,νTλS
−1
µ,νT
−1
λ =
1 µ(1− λ) 0 ν
(
1− λ−1
)
0 1 0 0
0 0 1 0
0 0 0 1
is not identically equal to the identity matrix. The condition λ = 1 implies than for every σ ∈ G
we have that σ
(
e
1
τ
)
= e
1
τ , i.e., e
1
τ ∈ C(τ), which is a contradiction. Thus the connected compo-
nent of the unit element of the differential Galois group of the system (3.3) is not Abelian. ■
As a consequence of Theorem 3.9, we have the following.
Corollary 3.10. The connected component (G2)
0 of the unit element of the differential Galois
group of the (LNVE)2 is not Abelian.
Proof. We always can put the independent variables in (LNVE)2 in such an order that the
variables w2, w1z1, y1w1, w2
1 stay in a block. For example, let the system (3.3) forms the
tail of the (LNVE)2. Then after the transformation t = 1/τ the (LNVE)2 admit such for-
mal Ψ̂(LNVE2)(τ) and actual Ψθ
(LNVE)2
(τ) fundamental matrix solutions at τ = 0 which contain
the matrices Ψ̂(τ) and Ψθ(τ) from Propositions 3.2 and 3.6, respectively, as right-hand lower
corner blocks. The differential Galois group G2 of the (LNVE)2 is a subgroup of GL13(C), so is
(G2)
0. With respect to the fundamental matrix solutions Ψ̂(LNVE2)(τ) and Ψθ
(LNVE)2
(τ) the con-
nected component of the unit element (G2)
0 of G2 is not Abelian since it has a proper subgroup,
which is not Abelian. ■
Combining Corollary 3.10 with the Morales–Ramis–Simó theory, we establish the main result
of this section.
Theorem 3.11. Assume that α1 = α2 = α3 = 0, α4 = 1, α0 = −α5, where α5 is arbitrary.
Then the Sasano system (1.1) is not integrable in the Liouville–Arnold sense by rational first
integrals.
4 Bäcklund transformations and generalization
In this section with the aid of the Bäcklund transformations of the Sasano system (1.1), we
extend the result of Theorem 3.11 to the entire orbit of the parameters αj , j = 0, . . . , 5, and
establish the main results of this paper.
Denote (∗) := (x, y, z, w, t;α0, α1, α2, α3, α4, α5). The action of the generators of the exten-
ded affine Weyl group W̃
(
D
(1)
5
)
on (∗) is defined as follows (see [27]):
s0(∗) =
(
x+
α0
y + t
, y, z, w, t;−α0, α1, α2 + α0, α3, α4, α5
)
,
s1(∗) =
(
x+
α1
y
, y, z, w, t;α0,−α1, α2 + α1, α3, α4, α5
)
,
s2(∗) =
(
x, y − α2
x− z
, z, w +
α2
x− z
, t;α0 + α2, α1 + α2,−α2, α3 + α2, α4, α5
)
,
s3(∗) =
(
x, y, z +
α3
w
,w, t;α0, α1, α2 + α3,−α3, α4 + α3, α5 + α3
)
,
s4(∗) =
(
x, y, z, w − α4
z − 1
, t;α0, α1, α2, α3 + α4,−α4, α5
)
,
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 21
s5(∗) =
(
x, y, z, w − α5
z
, t;α0, α1, α2, α3 + α5, α4,−α5
)
,
π1(∗) = (1− x,−y − t, 1− z,−w, t;α1, α0, α2, α3, α5, α4),
π2(∗) =
(
y + w + t
t
,−t(z − 1),
y + t
t
,−t(x− z),−t;α5, α4, α3, α2, α1, α0
)
,
π3(∗) = (1− x,−y, 1− z,−w,−t;α0, α1, α2, α3, α5, α4),
π4(∗) = (x, y + t, z, w,−t;α1, α0, α2, α3, α4, α5). (4.1)
In fact, the actions sj , 0 ≤ j ≤ 5 define a representation of the affine Weyl group W
(
D
(1)
5
)
.
Remark 4.1. When y = −t (resp. y = 0) is a particular solution of the Sasano system (1.1),
then the parameter α0 (resp. α1) must be equal to zero. So, when y = −t (resp. y = 0),
we consider the transformation s0 (resp. s1) as an identity transformation. Note that α1 = 0
does not imply y = 0 of necessity. For example, if α1 = 0, the function y = −t is a particular
solution of the system (1.1), provided that α0 = 0. Next, when z = 0 (resp. z = 1) is a particular
solution of (1.1), we find that α5 = 0 (resp. α4 = 0). This time we consider the transformation s5
(resp. s4) as an identity transformation. Note that α5 = 0 does not imply z = 0 of necessity.
For example, when α5 = 0, the function z = t is a particular solution of the system (1.1),
provided that α4 = −1 and y = − t
2 , w = 0 solve the same system. When w = 0, α3 = 0 we
consider the transformation s3 as an identity transformation. Finally, when x = z, α2 = 0, the
transformation s2 is considered as an identity transformation.
Denote by V := (α0, . . . , α5) = (−α, 0, 0, 0, 1, α) the vector of parameters corresponding
to the particular solution Sol := (x, y, z, w) =
(
α
t , 0,
α
t , 0
)
. In order to describe the orbit of the
vector V under the action of the group W̃
(
D
(1)
5
)
, we define using the ideas of Sasano the following
translation operators:
T1 = π1s5s3s2s1s0s2s3s5, T2 = π2T1π2, T3 = s1s4T1s4s1,
T4 = s2s3T3s3s2, T5 = s1T4s1, T6 = s3T3s3.
These operators act on the parameters as follows:
T1(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (0, 0, 0, 0, 1,−1),
T2(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (−1, 1, 0, 0, 0, 0),
T3(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (0, 0, 0, 1,−1,−1),
T4(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (1, 1,−1, 0, 0, 0),
T5(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (1,−1, 0, 0, 0, 0),
T6(α0, α1, . . . , α5) = (α0, α1, . . . , α5) + (0, 0, 1,−1, 0, 0).
Note that the particular solution Sol is transformed under the transformations Tj as follows:
T1(Sol) =
(
α
t
− 1
t− α+ 1
, 0,
α
t
− 1
t− α+ 1
, 0
)
,
T2(Sol) =
(
1− 1
α− t+ 1
,−1 +
α
α− t
, 1− 1
α− t+ 1
, 0
)
,
T3(Sol) = (1, 0, 1,−t+ α− 1),
T4(Sol) =
(
1,−t+ α− 1, 1− 1
t− α+ 1
, 0
)
,
T5(Sol) =
(
1− 1
t− α+ 1
,−t+ α− 1, 1− 1
t− α+ 1
, 0
)
,
22 Ts. Stoyanova
T6(Sol) =
(
1, 0, 1− 1
t− α+ 1
,−t+ α− 1
)
.
Note also that from Remark 4.1 it follows that all of the Bäcklund transformations (4.1) make
sense for the particular solution Sol.
Denote M1 = 1 − α0 − α1, M2 = α4 + α5. The next two lemmas describe the orbit of
the vector V under the group W̃
(
D
(1)
5
)
with generators (4.1).
Lemma 4.2. Assume that α /∈ Z. Let (x, y, z, t) =
(
α
t , 0,
α
t , 0
)
be a particular rational solution
of the Sasano system (1.1) with vector of parameters V . Then applying Bäcklund transforma-
tions (4.1) to this solution, we obtain rational solutions of (1.1) with parameters αj, 0 ≤ j ≤ 5,
which are either of the kind ±α + nj, nj ∈ Z or of the kind lj, lj ∈ Z in such a way that M1
and M2 are together of the kind ±α+mj, mj ∈ Z, j = 1, 2.
Proof. This lemma is proved by an induction on the numbers of the applied transforma-
tions (4.1) on the vector V and with the aid of above translation operators Tj , 1 ≤ j ≤ 6. ■
Remark 4.3. The conditions imposed on the parameters αj by Lemma 4.2 ensure that when
α /∈ Z, at least two of the new-obtained parameters αj are integer numbers and at least two of
them are not integer numbers.
With the next lemma, we specify the orbit of the vector V when α ∈ Z.
Lemma 4.4. Assume that α ∈ Z. Let (x, y, z, t) =
(
α
t , 0,
α
t , 0
)
be a particular rational solution
of the Sasano system (1.1) with vector of parameters V . Then applying Bäcklund transforma-
tions (4.1) to this solution we obtain rational solutions of (1.1), for which all of the parame-
ters αj, 0 ≤ j ≤ 5, are integer numbers in such a way that M1 and M2 are together either even
or odd integer.
Proof. This lemma is proved inductively. ■
Following [28], we define the symplectic transformations ri, 0 ≤ i ≤ 5, which correspond to
the symmetries si, 0 ≤ i ≤ 5, from (4.1)
r0(x, y, z, w) =
(
1
x
,−t− x(x(y + t) + α0), z, w
)
,
r1(x, y, z, w) =
(
1
x
,−x(xy + α1), z, w
)
,
r2(x, y, z, w) =
(
z − y(y(x− z)− α2),
1
y
, z, w + y − 1
y
)
,
r3(x, y, z, w) =
(
x, y,
1
z
,−z(zw + α3)
)
,
r4(x, y, z, w) =
(
x, y, 1− w(w(z − 1)− α4),
1
w
)
,
r5(x, y, z, w) =
(
x, y,−w(wz − α5),
1
w
)
.
The transformations πj , 1 ≤ j ≤ 5, from (4.1) are also canonical symplectic transformations since
dx ∧ dy + dz ∧ dw = dπj(x) ∧ dπj(y) + dπj(z) ∧ dπj(w).
Denote, in short, by ri, 0 ≤ i ≤ 5, and πj , 1 ≤ j ≤ 4, the image of the canonical coordi-
nates x, y, z, w under the action of ri and πj , respectively. Then the following important result
is an extension of [28, Theorem 4.1].
Non-Integrability of the Sasano System of Type D
(1)
5 and Stokes Phenomena 23
Theorem 4.5. There exists an unique polynomial Hamiltonian system of degree 4, which is
holomorphic in each coordinates ri, 0 ≤ i ≤ 5, and πj, 1 ≤ j ≤ 5. This system is invariant
under the extended Weyl group W̃
(
D
(1)
5
)
and coincides with the system (1.1).
Theorem 4.5 says that the transformations si, 0 ≤ i ≤ 5, and πj , 1 ≤ j ≤ 4, from (4.1) are
canonical transformations, which are rational on the functions x, y, z, w. Using this fact, we
establish the main results of this paper.
Theorem 4.6. Let α be an arbitrary complex parameter, which is not an integer. Assume that
the parameters αj are either of the kind ±α+nj or of the kind lj , nj , lj ∈ Z in such a way that M1
and M2 are together of the kind ±α + mi, mi ∈ Z, i = 1, 2. Then the Sasano system (1.1) is
not integrable in the Liouville–Arnold sense by rational first integrals.
Theorem 4.7. Assume that all of the parameters αj, 0 ≤ j ≤ 5, are integer in such a way
that M1 and M2 are together either even or odd integer. Then the Sasano system (1.1) is not
integrable in the Liouville–Arnold sense by rational first integrals.
Acknowledgments
The author is indebted to the referees for critical remarks and advices towards the improvement
of the paper. The author was partially supported by Grant KP-06-N 62/5 of the Bulgarian
Fund “Scientific research”.
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1 Introduction
2 Preliminaries
2.1 Non-integrability of Hamiltonian systems and differential Galois theory
2.2 Differential Galois group of a linear system of ordinary differential equations
3 Non-integrability for alpha_1=alpha_2=alpha_3=0, alpha_4=1, alpha_0=-alpha_5
4 Bäcklund transformations and generalization
References
|
| id | nasplib_isofts_kiev_ua-123456789-212871 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:56:45Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Stoyanova, Tsvetana 2026-02-13T13:48:53Z 2025 Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena. Tsvetana Stoyanova. SIGMA 21 (2025), 020, 24 pages 1815-0659 2020 Mathematics Subject Classification: 34M55; 37J30; 34M40; 37J65 arXiv:2306.07062 https://nasplib.isofts.kiev.ua/handle/123456789/212871 https://doi.org/10.3842/SIGMA.2025.020 In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type ⁽¹⁾ₗ, = 4, 5, 6, …. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group ˜(⁽¹⁾₅) and which we call the Sasano system of type ⁽¹⁾₅. We prove that one family of the Sasano system of type ⁽¹⁾₅ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result. The author is indebted to the referees for their critical remarks and advice towards the improvement of the paper. The author was partially supported by Grant KP-06-N 62/5 of the Bulgarian Fund “Scientific research”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena Article published earlier |
| spellingShingle | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena Stoyanova, Tsvetana |
| title | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena |
| title_full | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena |
| title_fullStr | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena |
| title_full_unstemmed | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena |
| title_short | Non-Integrability of the Sasano System of Type ⁽¹⁾₅ and Stokes Phenomena |
| title_sort | non-integrability of the sasano system of type ⁽¹⁾₅ and stokes phenomena |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212871 |
| work_keys_str_mv | AT stoyanovatsvetana nonintegrabilityofthesasanosystemoftype15andstokesphenomena |