Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice
This work aims to investigate the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra ⁽²⁾₄. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable...
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| description | This work aims to investigate the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra ⁽²⁾₄. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 087, 26 pages
Algebraic Complete Integrability
of the a
(2)
4 Toda Lattice
Bruce Lionnel LIETAP NDI a, Djagwa DEHAINSALA b and Joseph DONGHO a
a) University of Maroua, Faculty of Sciences, Department of Mathematics Computer Sciences,
P.O. Box 814, Maroua, Cameroon
E-mail: nbruce.lionnel@gmail.com, josephdongho@yahoo.fr
b) Department of Mathematics, Faculty of Exact and Applied Sciences, University of NDjamena,
1 route de Farcha, P.O. Box 1027, NDjamena, Chad
E-mail: djagwa73@gmail.com
Received April 25, 2024, in final form September 25, 2024; Published online October 05, 2024
https://doi.org/10.3842/SIGMA.2024.087
Abstract. The aim of this work is focused on the investigation of the algebraic complete
integrability of the Toda lattice associated with the twisted affine Lie algebra a
(2)
4 . First,
we prove that the generic fiber of the momentum map for this system is an affine part of an
abelian surface. Second, we show that the flows of integrable vector fields on this surface
are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed
geometric description of these abelian surfaces and the divisor at infinity.
Key words: Toda lattice; integrable system; algebraic integrability; abelian surface
2020 Mathematics Subject Classification: 34G20; 34M55; 37J35
1 Introduction
The study of integrable Hamiltonian systems has been motivated by several factors, including the
development of powerful and beautiful mathematical theories and the application of integration
concepts to various physical, biological and chemical systems. However, it remains challenging
to describe or recognize integrable Hamiltonian systems with ease, as they are exceptional cases.
The Korteweg–de Vries equation has generated numerous new ideas in the field of completely
integrable Hamiltonian systems, leading to unexpected connections between mechanics, spectral
theory, Lie algebra theory, algebraic geometry, and even differential geometry. Some interesting
integrable systems also appear as coverings of algebraic complete integrable systems. These
systems are sometimes also called algebraic complete integrable.
An algebraic complete integrable system can be linearized on a complex torus, and its invari-
ant functions (often called first integrals or constants) are polynomial maps and their restrictions
to an invariant complex variety are meromorphic functions on a complex abelian variety. The
fluxes generated by the constants of motion are straight lines in this complex abelian variety.
Several nonlinear completely integrable systems were known in the 19th century, among them
the geodesic flow on the ellipsoid, Neumann’s system or the Kowalevski top. The Toda lattice,
introduced by Morikazu Toda in 1967, is a simple model for a one-dimensional crystal in solid-
state physics. It is famous because it is one of the first integrable systems for which a Lax pair
was discovered by Flaschka. The classical Toda lattice is a system of particles with unit mass,
connected by exponential springs. Its equations of motion derived from the Hamiltonian
H =
1
2
n∑
j=1
p2j +
n−1∑
j=1
eqj−qj+1 , (1.1)
mailto:nbruce.lionnel@gmail.com
mailto:josephdongho@yahoo.fr
mailto:djagwa73@gmail.com
https://doi.org/10.3842/SIGMA.2024.087
2 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
where qj is the position of the j-th particle and pj is its amount of movement. This type of
Hamiltonian was considered first by Morikazu Toda [12, 13]. The equation (1.1) is known as
the finite classic non-periodic Toda lattice to distinguish other versions of various forms of the
system. The periodic version of (1.1) is given by
H =
1
2
n∑
j=1
p2j +
n∑
j=1
eqj−qj+1 , qn+1 = q1,
where the equations of motion are given by
ṗj = −∂H
∂qj
= e(qj−1−qj) − e(qj−qj+1), q̇j =
∂H
∂pj
= pj , 1 ≤ j ≤ n.
The integrability of the periodic Toda lattice was established by Henon [10] and Flaschka [8]
using the Lax pairs method. In 1976, Bogoyavlensky [5] introduced a generalization of the
classical Toda periodic lattice to arbitrary Lie algebras.
Adler, van Moerbeke and Vanhaecke in [4] give the explicit Hamiltonians for the periodic
Toda lattices that involve precisely three (connected) particles. In two-dimensional, there are
precisely six cases of them, going with the extended root systems a
(1)
2 , a
(2)
4 , c
(1)
2 , d
(2)
3 , g
(1)
2 and d
(3)
4 .
They prove in [1, 4] that the case a
(1)
2 is algebraic completely integrable. In his case, Dehainsala
prove in [6] that the two cases c
(1)
2 and d
(2)
3 are algebraic complete integrable.
In this work, we consider that a
(2)
4 is a two-dimensional integrable system. This system
satisfies the linearization criterion [4, Theorem 6.41]. We prove that this system is an algebraic
completely integrable in the Adler–van Moerbeke sense.
To prove this, with respect to the complex Liouville theorem, firstly, by the indicial locus and
the Kowalevski matrix, we have shown that our system has three distinct families of homogeneous
Laurent solutions with weights depending on four free parameters and find the Zariski open set Ω
and the fiber Fc, c ∈ Ω. Secondly, with these Laurent solutions, we determined the Painlevé divi-
sor and their arithmetic genus. We have obtain three divisors, for c ∈ Ω, the Painlevé divisor Γ
(0)
c
is a smooth genus three hyperelliptic curve, the Painlevé divisor Γ
(1)
c is a smooth genus four curve
and the Painlevé divisor Γ
(2)
c is a smooth genus two hyperelliptic curve. Thirdly, to compact
the fiber, we determined the projective space to embedding the divisor at infinity to find the
singularities and the intersection between the different curves. We have obtain twenty-five (25)
functions which forms the basis of the projective space and determined the intersections points.
To end our prove, we show that the vector field (φc)∗V1 extends to an holomorphic vector fields
on P24. To show that the vector field (φc)∗V1 is holomorphic on two chart of P24, we have estab-
lished that this vector field can be written as a quadratic vector field in two appropriate chart.
This paper is organized as follows. In Section 2, we review the basic notions of algebraic
integrability in sense of Adler–van Moerbeke. Section 3 contains the main part of the paper,
we verify that the a
(2)
4 Toda lattice is Liouville integrable, we do the Painlevé analysis of the
system. This analysis shows that our integrable system admits three principal balances, i.e.,
three families of Laurent solutions depending on the maximal number of free parameters,four
in our case. Thus, by confining each family of Laurent solutions to the invariant manifolds we
calculate the Painlevé divisors associated to these principal balances and, for c ∈ Ω, we give an
explicit embedding of the invariant manifold Fc in the projective space P24. Finally, in Section 4,
we determine the holomorphic differentials forms on the abelian surface.
2 Preliminaries
In this section, we also recall some basics notions. For more comprehension just read the book [4].
Let v = (v1, . . . , vn) be a collection of positive integers without a common divisor. Such a v
is called a weight vector.
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 3
Definition 2.1 ([4]). A polynomial f ∈ F(Cn) := C[x1, . . . , xn] is a weight homogeneous polyno-
mial of weight k (with respect to v) if f(tv1x1, . . . , t
vnxn) = tkf(x1, . . . , xn), ∀ (x1, . . . , xn) ∈ Cn
and t ∈ C.
When the weight of f is k ∈ N, we denote
F (k) := {F ∈ F(Cn) | ϖ(F ) = k}.
Definition 2.2 ([4]). A polynomial vector field on Cn
ẋ1 = f1(x1, . . . , xn),
· · · · · · · · · · · · · · · · · · ·
ẋn = fn(x1, . . . , xn)
(2.1)
is called a weight homogeneous vector field of weight k (with respect to v) if each of the poly-
nomials f1, . . . , fn is weight homogeneous (with respect to v) and if ϖ(fi) = vi + k = ϖ(xi) + k
for i = 1, . . . , n.
According to [4, Proposition 7.6], if (2.1) is a weight homogeneous vector field, then Laurent
solutions have the form
xi(t) =
1
tvi
∞∑
k=0
x
(k)
i tk, i = 1, . . . , n, with x(0) =
(
x
(0)
1 , . . . , x(0)n
)
̸= 0,
are called weight homogeneous Laurent solution. We will say that a formal Laurent solution is
a principal balance if it depends on n− 1 free parameters; otherwise, it will be called the lower
balance.
The positive integers vi being the weights of the phase variables, then the leading coef-
ficients x
(0)
i satisfy the nonlinear algebraic equations
vix
(0)
i + fi
(
x
(0)
1 , x
(0)
2 , . . . , x(0)n
)
= 0, i = 1, . . . , n, (2.2)
while the other coefficients x
(k)
i , k ≥ 1, satisfy(
k Idn−K
(
x(0)
))
x(k) = P (k),
where
x(k) =
(
x
(k)
1 , x
(k)
2 , . . . , x(k)n
)T
, P (k) =
(
P
(k)
1 , P
(k)
2 , . . . , P (k)
n
)T
and each P
(k)
i is a polynomial which depends only on the variables x
(l)
1 , x
(l)
2 , . . . , x
(l)
n with
0 ≤ l ≤ k. The matrix K of order n is defined by Kij =
∂fi
∂xj
+ viδij is called the Kowalevski
matrix.
Let Λ be a discrete subgroup of rank k of Cn. The quotient group Cn/Λ has a smooth
complex manifold structure induced by the projection π : Cn −→ Cn/Λ. This complex manifold
is compact if and only if k = 2n. In this case, Tn = Cn/Λ is a complex torus. If such a complex
torus is also an algebraic variety, i.e., admits a projective embedding, we then say that it is an
abelian variety. If k = 2n = 4, we say that it is an abelian surface.
To define an algebraic complete integrable system, we need the following definitions.
Definition 2.3 ([4]). A fiber of F is a common level set of the functions Fi. The fiber of F that
passes through m ∈M will be denoted by
Fm = {p ∈M | Fi(p) = Fi(m), ∀i ∈ {1, . . . , s}}.
4 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
For c ∈ Rs (resp. Cs), we will note the fiber F−1(c) above c by Fc. So, we have
Fm = F−1(F(m)) = FF(m) ∀m ∈M.
The set of regular values of F is a residual subset (hence a dense subset) of Rs (resp. Cs). By the
inverse function theorem, the fiber Fc over each regular value c that lies in the image of F is
non-singular. Hence, when F = (F1, . . . , Fs) is involutive, the Hamiltonian vector fields XFi ,
1 ≤ i ≤ s, commute and for any pointm they are tangent to the non-singular affine part of F(m).
Definition 2.4. An abelian variety is a complex torus Cr/Λ (Λ a lattice in Cr) which is pro-
jective, which means that it admits an embedding in a project space PN .
An abelian variety Tr will be called an irreducible abelian variety, when Tr does not contain
any abelian subvariety, otherwise it will be called a reducible abelian variety.
Definition 2.5. Let (M, {·, ·},F) be a complex integrable system, where M is a non-singular
affine variety and where F = (F1, . . . , Fs). We say that (M, {·, ·},F) is an algebraic completely
integrable system if for generic c ∈ Cs the fiber Fc is an affine part of an abelian variety
and if the Hamiltonian vector fields XFi are translation invariant, when restricted to these
fibers. In the particular case in which M is an affine space Cn, we will call (Cn, {·, ·},F)
a polynomial algebraic complete integrable system. When the generic abelian variety of the
algebraic complete integrable system is irreducible, we speak of an irreducible algebraic complete
integrable system.
An integrable system is said to be algebraic completely integrable if the fibers of the momen-
tum map are affine parts of abelian varieties and the integrable fields are linear.
The following theorem gives a necessary condition for the algebraic integrability of an inte-
grable system. It is inspired by the Kowalevski work [11], and is based on the fact that the
phase space of an algebraic complete integrable system admits a partial compactification on
which the integrable vector fields extend into complete vector fields. This means that each
of the integrable vector fields of an irreducible algebraic complete integrable system on Cn
admits one or several families of Laurent solutions (called balances), which will lead to a nec-
essary condition for algebraic complete integrability, which we call the Kowalevski–Painlevé
criterion.
Theorem 2.6 (Kowalevski–Painlevé criterion [4]). Let (Cn, {·, ·},F) be an irreducible polyno-
mial algebraic complete integrable system, where F = (F1, . . . , Fs) is a family of polynomials and
(x1, . . . , xn) is a system of linear coordinates on Cn. Let V be any one of the integrable vector
fields XF1 , . . . ,XFs. For each 1 ≤ i ≤ n such that xi is not constant along the integral curve
of V, i.e., ẋi := V[xi] ̸= 0, there exists a principal balance x(t) = (x1(t), . . . , xn(t)), depending
on n− 1 free parameters for which xi(t) has a pole.
Let A an affine variety, φ : A −→ PN a regular map, let φ(A) be the closure of the image
of A in PN .
The following theorem gives the sufficient conditions to be satisfied by the fibers of the
momentum map of an integrable system to be algebraic complete integrable.
Theorem 2.7 (complex Liouville theorem [4]). Let A ∈ Cs be a non-singular affine variety of
dimension r which supports r holomorphic vector fields V1, . . . ,Vr and let φ : A −→ CN ⊂ PN be
a regular map; here CN ⊂ PN is the usual inclusion of CN as the complement of a hyperplane H
in PN . We define ∆ := φ(A) \ φ(A) and we decompose the analytic subset ∆ as ∆ = ∆′ ∪∆′′,
where ∆′ is the union of the irreducible components of ∆ of dimension r−1 and ∆′′ is the union
of the other irreducible components of ∆. The following conditions are assumed to be verified:
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 5
1. φ : A −→ CN is an isomorphic embedding.
2. The vector fields commute pairwise, [Vi,Vj ] = 0 for 1 ≤ i, j ≤ r.
3. At every point m ∈ A, the vector fields V1, . . . ,Vr are independent.
4. The vector field φ∗V1 extends to a vector field V1 which is holomorphic on a neighborhood
of ∆′ in PN .
5. The integral curves of V1 that start at points m ∈ ∆′ go immediately into φ(A).
Then φ(A) is an abelian variety of dimension r and ∆′′ = ∅, so that φ(A) = φ(A) ∪∆′. More-
over, the vector fields φ∗V1, . . . , φ∗Vr extends to holomorphic vector fields on φ(A).
3 Algebraic integrability of the a
(2)
4 Toda lattice
The aims of this section is to prove, by following the strategy developed by Adler, van Moerbeke
and Vanhaecke in [4, Section 9.4] to establish the algebraic integrability of the a
(1)
2 Toda lattice,
the algebraic complete integrability of the a
(2)
4 Toda lattice.
3.1 Liouville integrability of the a
(2)
4 Toda lattice
The differential equations of the periodic a
(2)
4 Toda lattice are given on the five dimensions
hyperplane H =
{
(x0, x1, x2, y0, y1, y2) ∈ C6 | y0 + 2y1 + 2y2 = 0
}
of C6 by{
ẋ = x.y,
ẏ = Ax,
where x = (x0, x1, x2)
T, y = (y0, y1, y2)
T and A is the Cartan matrix of the twisted affine Lie
algebra a
(2)
4 given in [4] by 2 −2 0
−1 2 −2
0 −1 2
and ε = (1, 2, 2)T is the normalized null vector of AT. The equations of motion of the a
(2)
4 Toda
lattice are given in [4] by
ẋ0 = x0y0, ẏ0 = 2x0 − 2x1,
ẋ1 = x1y1, ẏ1 = −x0 + 2x1 − 2x2,
ẋ2 = x2y2, ẏ2 = −x1 + 2x2. (3.1)
We denote by V1 the vector field defined by the above differential equations (3.1). Then V1 is
the Hamiltonian vector field, with Hamiltonian function F2 = y20 + 4y22 − 4x0 − 8x1 − 16x2 with
respect to the Poisson structure {·, ·} defined by the following skew-symmetric matrix:
J =
1
8
0 0 0 4x0 −2x0 0
0 0 0 −2x1 2x1 −x1
0 0 0 0 −x2 x2
−4x0 2x1 0 0 0 0
2x0 −2x1 x2 0 0 0
0 x1 −x2 0 0 0
. (3.2)
This Poisson structure is given on C6; the function F0 = y0 + 2y1 + 2y2 is a Casimir, so that
the hyperplane H is a Poisson subvariety. The rank of this Poisson structure {·, ·} is 0 on the
6 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
three-dimensional subspace {x0 = x1 = x2 = 0}; the rank is 2 on the three four-dimensional
subspaces: {x0 = x1 = 0}, {x0 = x2 = 0} and {x1 = x2 = 0}. Thus, for all points of H except
the four subspaces above the rank is 4. The vector field V1 admits also the following three
constants of motion:
F1 = x0x
2
1x
2
2,
F2 = y20 + 4y22 − 4x0 − 8x1 − 16x2,
F3 =
(
y20 − 4x0
)(
y22 − 4x2
)
− 4x1(y0y2 − 4x2 − x1). (3.3)
F1 is a Casimir for {·, ·}, and the function F3 generates a second Hamiltonian vector field V2,
which commutes with V1, given by the differential equations
x′0 = x0y2(y0y2 − 2x1)− 4x0x2y0,
x′1 = −x1y1y2(y1 + y2)− x21y1 + x1(x0y2 + 2x2y0),
x′2 = x2(y1 + y2)((y1 + y2)y2 + x1) + x0x2y0,
y′0 = 2
(
2x1x2 + x0y
2
2
)
+ x1(2x1 − y0y2)− 8x0x2,
y′1 = −x0y22 + 2x2(3x0 − x1) + y0y2(x1 + x2)− 2x21 + x2y0y1,
y′2 = x1y2(y1 + y2) + x21 − x2(y1 + y2)− 2x2x0. (3.4)
Hence, the system (3.1) is completely integrable in the Liouville sense. It can be written as
a Hamiltonian vector fields
ż = J
∂H
∂z
, z = (z1, . . . , z6)
T = (x0, x1, x2, y0, y1, y2)
T,
where H = F2. The Hamiltonian structure is defined by the following Poisson bracket:
{F,H} =
〈
∂F
∂z
, J
∂H
∂z
〉
=
6∑
i,k=1
Jik
∂F
∂zi
∂H
∂zk
,
where ∂H
∂z =
(
∂H
∂x0
, ∂H∂x1
, ∂H∂x2
, ∂H∂y0 ,
∂H
∂y1
, ∂H∂y2
)T
and J is an antisymmetric matrix.
The vector field V2 admits the same constants of motion (3.3) and is in involution with V1
therefore {F2, F3} = 0. The involution σ defined on C6 by
σ(x0, x1, x2, y0, y1, y2) = (x0, x1, x2,−y0,−y1,−y2)
preserves the constants of motion F1, F2 and F3, hence leave the fibers of the momentum map F
invariant. This involution can be restricts to the hyperplane H.
Let F = (F1, F2, F3) : H → C3 be the momentum map; functions Fi being two by two in
involution, F is involutive. The Jacobian matrix of F is given by
Jac :=
x21x
2
2 2x0x1x
2
2 2x0x
2
1x2 0 0
−4 −8 −16 2y0 8y2
16x2 − 4y22 ⋆ −4y20 ♦ △
with
⋆ = −4y0y2 + 16x2 + 8x1,
♦ = 2y0
(
y22 − 4x2
)
− 4x1y2,
△= 2y2
(
y20 − 4x0
)
− 4x1y0.
Let p0
(
1, 1, 1, 1,−3
2 , 1
)
be a point of H. The Jacobian matrix of F at p0 is given by
Jac(p0) =
1 2 2 0 0
−4 −8 −16 2 8
12 20 28 −10 −10
.
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 7
The rank of matrix Jac(p0) is 3, so the differentials dFi, i = 1, 2, 3, are independent at p0 and
since the functions Fi are polynomials, F is independent on a dense open subset UF ofH. We can
deduce that (H, {·, ·},F) is an integrable system in the sense of Liouville.
Let S be the set of points in H where the determinants of all 3× 3 minors of the matrix Jac
cancel. By direct computation, we prove that S is the union of following subvarieties:
S1 := {x1 = 0}, S2 := {x2 = 0}, S3 :=
{
x1 = 2y22, x0 = 4x2, y0 = 4y2
}
,
S4 :=
{
y0 = y2 = 0, x1 =
2x22 + x0x2 + α1
x0
, x1 =
2x22 + x0x2 − α1
x0
}
,
S5 :=
{
x0 =
1
4
β, x2 =
1
2
y2
(
4x1y2 − 2x1y0 − 2y0y
2
2 + y20y2
)
y0(−4y2 + y0)
}
with
α1 =
√
4x42 + 4x32x0 − 7x20x
2
2 + 2x2x30,
β =
8x1y0y2 − 8x1y
2
2 + 4y0y
3
2 − 4y20y
2
2 − 2x1y
2
0 + y30y2
y2(−4y2 + y0)
.
The images under F of S1 and S2 are contained in the subset F1 = c1 = 0. By substituting
x1 = 1
8y
2
0, x2 = 1
4x0 and y0 = 4y2 in the three constants of motion Fi = ci, i = 1, 2, 3, by direct
computation with Maple, we obtain
c1 =
1
4
x30y
4
2, c2 = 4y22 − 8x0, c3 = 4x0
(
−3y22 + x0
)
.
Using Fi = ci, i = 1, 2, 3, and eliminating the variables xi, yi, i = 1, 2, 3, in the above expresions,
we deduce that the image under F of S3 is contained in the subset
256
(
3200000c21 + 2000c23c2c1 − 225c3c
3
2c1 + c53
)
+ 1728c52c1 − 32c43c
2
2 + c33c
4
2 = 0,
and the set of regular values of the momentum map F is the Zariski open subset Ω defined by
Ω =
{
c = (c1, c2, c3) ∈ C3 | c1 ̸= 0 and
256
(
3200000c21 + 2000c23c2c1 − 225c3c
3
2c1 + c53
)
+ 1728c52c1 − 32c43c
2
2 + c33c
4
2 ̸= 0
}
.
At a generic point c = (c1, c2, c3) ∈ C3, the fiber on c ∈ Ω of F is therefore
Fc := F−1(c) =
3⋂
i=1
{m ∈ H | Fi(m) = ci}.
Hence, we have the following result which prove that a
(2)
4 Toda lattice is a completely integrable
system in the Liouville sense.
Proposition 3.1. For c ∈ Ω, the fiber Fc over c of the momentum F is a smooth affine variety
of dimension 2 and the rank of the Poisson structure (3.2) is maximal and equal to 4 at each
point of Fc; moreover, the vector fields V1 and V2 are independent at each point of the fiber Fc.
Proposition 3.2. (H, {·, ·},F) is a completely integrable system describing the a
(2)
4 Toda lattice,
where F = (F1, F2, F3) and {·, ·} are given respectively by (3.3) and (3.2) with commuting vector
fields (3.1) and (3.4).
8 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
3.2 Algebraic integrability of a
(2)
4 Toda lattice
To show that the a
(2)
4 Toda lattice is algebraically completely integrable, we show that for all
c = (c1, c2, c3) ∈ Ω, the fiber Fc is the affine part of an abelian surface on which the two vector
fields V1 and V2 are linear [6]. For that it is necessary to verify that Fc satisfies the conditions
of the Liouville complex theorem (Theorem 2.7).
Observe that the vector field V1 is homogeneous with respect to the following weights:
ϖ(x0, x1, x2, y0, y1, y2) = (2, 2, 2, 1, 1, 1).
We can also verify that the constants of motion Fi are weight homogeneous and they have the
following weights: ϖ(F1, F2, F3) = (10, 2, 4).
3.2.1 Laurent solutions
According to [4], the vector fields of an algebraic complete integrable system have good properties
at infinity; after all, a linear vector field on a complex torus is the same along the divisor, which
will happen to be absent in phase space, as on the rest of the torus. In fact, since every
holomorphic function on a complex torus can be written as a quotient of theta functions, the
integral curves (solutions) to any of the vector fields of an algebraic complete integrable system
can be written as a quotient of holomorphic functions. Intuitively speaking this means that we
can consider not only Taylor solutions to the differential equations that describe these vector
fields but also Laurent solutions, which will correspond to initial conditions at infinity (precisely:
on the divisor that needs to be adjoined to the fibers of the momentummap to complete them into
abelian varieties). Moreover, these Laurent solutions must depend on dimH−1 free parameters,
which corresponds to the freedom of choice of the initial condition at infinity.
Let us look for weights homogeneous Laurent solutions associated to the vector field V1.
The solution form of vector field V1 is
xi(t) =
1
t2
∞∑
k=0
x
(k)
i tk and yi(t) =
1
t
∞∑
k=0
y
(k)
i tk, i = 0, 1, 2,
that is to say that ϖ(xi) = 2ϖ(yi) = 2 for i = 0, 1,2. By substituting these solutions into the
differential equations (3.1) associated with the vector field V1, the indicial locus is the subset
of H given according to (2.2) by
0 = x
(0)
0
(
2 + y
(0)
0
)
,
0 = x
(0)
1
(
2 + y
(0)
1
)
,
0 = x
(0)
2
(
2 + y
(0)
2
)
,
0 = y
(0)
0 + 2x
(0)
0 − 2x
(0)
1 ,
0 = y
(0)
1 − x
(0)
0 + 2x
(0)
1 − 2x
(0)
2 ,
0 = y
(0)
2 − x
(0)
1 + 2x
(0)
2 .
These equations are easily solved and they yield the following (non-zero) solutions:
m0 = (1, 0, 0,−2, 1, 0), m3 = (0, 4, 3, 8,−2,−2),
m1 = (0, 1, 0, 2,−2, 1), m4 = (1, 0, 1,−2, 3,−2),
m2 = (0, 0, 1, 0, 2,−2), m5 = (4, 3, 0,−2,−2, 3).
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 9
The Kowalevski matrix at an arbitrary point
(
x(0), y(0)
)
, solution of the indicial equation is
given by
K
(
x(0), y(0)
)
=
2 + y
(0)
0 0 0 x
(0)
0 0
0 2− 1
2y
(0)
0 − y
(0)
2 0 −1
2x
(0)
1 −x(0)1
0 0 2 + y
(0)
2 0 x
(0)
2
2 −2 0 1 0
0 −1 2 0 1
.
Lemma 3.3. The system of differential equation (3.1) of the vector field V1 has three distinct
families of homogeneous Laurent solutions with weights depending on four (dimH − 1) free
parameters.
Proof. The characteristic polynomial of the Kowalevski matrix at m0 is given by
X (λ;m0) = (λ− 1)(λ− 3)(λ+ 1)(λ− 2)2.
It admits 4 non-negative eigenvalues which leads to a Laurent solution depending on four free
parameters, whose the five leading terms (going with steps 1, 2, 2, 3, respectively, are denoted
by a, c, d and e). The first four terms of this balance which we note x(t,m0):
x0(t,m0) =
1
t2
+ c− 1
2
ed+O
(
t2
)
,
x1(t,m0) = et+O
(
t2
)
,
x2(t,m0) = d− adt+O
(
t2
)
,
y0(t,m0) = −2
t
+ 2ct− 3
2
et2 +O
(
t3
)
,
y1(t,m0) =
1
t
+ a− (c+ 2d)t+
(
ad+
5
4
e
)
t2 +O
(
t3
)
,
y2(t,m0) = −a+ 2dt−
(
ad+
1
2
e
)
t2 +O
(
t3
)
. (3.5)
The characteristic polynomial of the Kowalevski matrix at m1 is given by
X (λ;m1) = (λ− 4)(λ− 1)(λ− 2)(λ− 3)(λ+ 1).
It admits 4 non-negative eigenvalues which leads to a Laurent solution depending on four free
parameters, whose the five leading terms (going with steps 1, 2, 3, 4, respectively, are denoted
by a, c, d and e). The first four terms of this balance which we note x(t,m1):
x0(t,m1) = et2 +O
(
t3
)
,
x1(t,m1) =
1
t2
+ c− 1
2
dt+
1
10
(
6c2 + ad− e
)
t2 +O
(
t3
)
,
x2(t,m1) = dt− 1
2
adt2 +O
(
t3
)
,
y0(t,m1) =
2
t
+ a− 2ct+
1
2
dt2 +
1
15
(
11e− 6c2 − ad
)
t3 +O
(
t4
)
,
y1(t,m1) = −2
t
+ 2ct− 3
2
dt2 +
2
5
(
c2 + ad− e
)
t3 +O
(
t4
)
,
y2(t,m1) =
1
t
− 1
2
a− ct+
5
4
dt2 +
1
30
(
e− 6c2 − 11ad
)
t3 +O
(
t4
)
. (3.6)
10 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
The characteristic polynomial of the Kowalevski matrix at m2 is given by
X (λ;m2) = (λ+ 1)(λ− 4)(λ− 1)(λ− 2)2.
It admits 4 non-negative eigenvalues which leads to a Laurent solution depending on four free
parameters, whose the five leading terms (going with steps 1, 2, 2, 4, respectively, are denoted
by a, c, d and e). The first four terms of this balance which we note x(t,m2):
x0(t,m2) = c+ act+
1
2
c
(
2c+ a2
)
t2 +O
(
t3
)
,
x1(t,m2) = et2 +O
(
t3
)
,
x2(t,m2) =
1
t2
+ d+
1
10
(
6d2 − e
)
t2 +O
(
t3
)
,
y0(t,m2) = a+ 2ct+ act2 +
1
3
(
2c2 + a2c− 2e
)
t3 +O
(
t4
)
,
y1(t,m2) =
2
t
− 1
2
a− (c+ 2d)t− 1
2
act2 − 1
30
(
10c2 + 5a2c+ 12d2 − 22e
)
t3 +O
(
t4
)
, (3.7)
y2(t,m2) = −2
t
+ 2dt+
2
5
(
d2 − e
)
t3 +O
(
t4
)
. ■
According to Adler and van Moerbeke [3], since the Toda problem is algebraic integrable,
[2, Theorem 1] implies the existence of a coherent tree of Laurent solutions, satisfying all the
conditions described in [4]. The existence of a coherent tree of Laurent solutions is crucial for
obtaining the complete structure of the divisor D at infinity in [3]. In this work, we do not
use a coherent tree of Laurent solutions, but only the so-called principal balances depending
on 4 free parameters corresponding to m0, m1, m2 and that m3, m4, m5 correspond to lower
balances depending on 3 free parameters.
3.2.2 Painlevé divisors of a
(2)
4 Toda lattice
We now search the formal Painlevé divisors, i.e., the algebraic curves defined by the three
different principal balances x(t,mi)i=0,1,2, confined to a fixed affine invariant surface Fc, c ∈ Ω.
We have the following assertion.
Proposition 3.4. For the Laurent solution x(t;m0) restricted to the invariant surface, for
c ∈ Ω, the Painlevé divisor Γ
(0)
c is a smooth genus three hyperelliptic curve. It is given by
Γ(0)
c : 16d2a8 −
(
256d3 + 8d2c2
)
a6 +
(
1536d2 + 96dc2 + 8c3 + c22
)
d2a4
−
((
8
(
8c3 + 48dc2 + c22 + 512d2
)
d+ 2c2c3
)
d2 + 64c1
)
a2
+
(
8d
(
c2c3 + 16dc3 + 64d2c2 + 512d3 + 2dc22
)
+ c23
)
d2 = 0.
It is completed in a Riemann surface, denoted Γ
(0)
c by adding 8 points to infinity.
Proof. Consider Laurent’s solution x(t;m0) in (3.5). When it is substituted in the equations
Fi = ci, i = 1, 2, 3, where ci = (c1, c2, c3) ∈ Ω, we get
c1 = e2d2, c2 = 4a2 − 12c− 16d, c3 = −12ca2 + 48cd− 8ae.
By eliminating the parameters c and e, we obtain an algebraic relation between a and d which
is the equation of an affine curve in C2 defined by
Γ(0)
c : 16d2a8 −
(
256d3 + 8d2c2
)
a6 +
(
1536d2 + 96dc2 + 8c3 + c22
)
d2a4
−
((
8
(
8c3 + 48dc2 + c22 + 512d2
)
d+ 2c2c3
)
d2 + 64c1
)
a2
+
(
8d
(
c2c3 + 16dc3 + 64d2c2 + 512d3 + 2dc22
)
+ c23
)
d2 = 0.
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 11
The affine curve Γ
(0)
c is smooth for c ∈ Ω. It is completed in a Riemann surface, denoted Γ
(0)
c
by adding 8 points to infinity denoted by ∞ϵ, ∞ϵ1ϵ2 and ∞3
ϵ3 with ϵ2 = ϵ21 = ϵ22 = ϵ23 = 1
and δ =
√
c22 − 16c3. A neighborhood of each of these points is described according to a local
parameter ς by
∞ϵ : a = ς−1, d =
1
2
ϵc2
√
c1ς
5 + 2ϵ
√
c1ς
3 +O
(
ς5
)
, (3.8)
∞ϵ1ϵ2 : a = ς−1, d =
1
4 +
(
− 1
32c2 +
1
32ϵ1δ
)
+ 8ϵ2ς
3
√
c1
c22−16c3
ς2
+O(ς), (3.9)
∞3
ϵ3 : a = ς, d = −256c1c2ς
2
c33
+ 8ϵ3ς
√
c1
c23
+O
(
ς3
)
, (3.10)
This completes the proof of Proposition 3.4. ■
Proposition 3.5. For the Laurent solution x(t;m1) restricted to the invariant surface, for
c ∈ Ω, the Painlevé divisor Γ
(1)
c is a smooth genus four curve. It is given by
Γ(1)
c : 256ad3 −
((
4a2 − c2
)2 − 16c3
)
d2 + 64c1 = 0.
It is completed in a Riemann surface, denoted Γ
(1)
c by adding 4 points to infinity.
Proof. Consider Laurent’s solution x(t;m1) in (3.6). When it was substituted in the equations
Fi = ci, i = 1, 2, 3, where ci = (c1, c2, c3) ∈ Ω, we obtain
c1 = ed2, c2 = 2a2 − 24c, c3 =
1
4
a4 + 6a2c− 16ad+ 36c2 − 4e
by eliminating the parameters c and e, we obtain an algebraic relation between a and d which
is the equation of an affine curve in C2 defined by
Γ(1)
c : 256ad3 −
((
4a2 − c2
)2 − 16c3
)
d2 + 64c1 = 0.
The affine curve Γ
(1)
c is smooth for c ∈ Ω. Indeed, let
g(a, d) = 256ad3 −
((
4a2 − c2
)2 − 16c3
)
d2 + 64c1,
we have
∂g
∂a
(a, d) = 16d2
(
4a3 − ac2 − 16d
)
,
∂g
∂d
(a, d) = 2d
(
16a4 − 8a2c2 − 384ad+ c22 − 16c3
)
.
Thus, a point (a, d) is singular for the affine curve Γ
(1)
c if
g(a, d) =
∂g
∂a
(a, d) =
∂g
∂d
(a, d) = 0
as d ̸= 0 since c1 ̸= 0, then
4a3 − ac2 − 16d = 0,
16a4 − 8a2c2 − 384ad+ c22 − 16c3 = 0,
−256ad3 +
(
16a4 − 8a2c2 − 384ad+ c22 − 16c3
)
d2 − 64c1 = 0,
12 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
⇔
4a3 − ac2 − 16d = 0,
16a4 − 8a2c2 − 384ad+ c22 − 16c3 = 0,
2ad3 − c1 = 0.
By expressing a as a function of d in the last equation of the system and substituting the
expression into both first equations, we obtain
a =
c1
2d3
, −c31 + c1c2d
6 + 32d10 = 0,
c1 − 2c21c2d
6 + c22d
12 − 192c1d
10 − 16c3d
12 = 0.
The resultant of these two last polynomials in d is given by the following polynomial in terms
of c1, c2 and c3 up to a constant
c361
(
819200000c21 + 512000c23c2c1 − 57600c3c
3
2c1 − 32c43c
2
2 + c33c
4
2 + 256c53 + 1728c52c1
)2
.
This expression is not zero for c ∈ Ω. We deduce that Γ
(1)
c is a smooth affine curve for c ∈ Ω.
It is complete into a Riemann surface, denoted Γ
(1)
c by adding four points at infinity denoted
by ∞1, ∞2 and ∞ϵ. A neighborhood of each of these points is described according to a local
parameter ς by
∞ϵ : d = ε
√
4
√
c1ς
2, a =
( c22
128 − c3
4
)
ς5 + c2ς3
8 + ς
ς2
+O
(
ς3
)
, (3.11)
∞1 : d = ς−1, a =
(
c22
256
− c3
16
)
ς +O
(
ς2
)
, (3.12)
∞2 : d =
1
16ς3
, a =
1
3
(
16c3 − c22
)
ς6 + 8c2
3 ς
4 + 16ς2
16ς3
+O
(
ς3
)
, (3.13)
This completes the proof of Proposition 3.5. ■
Proposition 3.6. For the Laurent solution x(t;m2) restricted to the invariant surface, for
c ∈ Ω, the Painlevé divisor Γ
(2)
c is a smooth genus two hyperelliptic curve. It is given by
Γ(2)
c : e4a4 −
(
8c1 + c2e
2
)
a2e2 − 64e5 + 4e2c1c2 + 4c3e
4 + 16c21 = 0.
It is completed in a Riemann surface, which is a double covering of P1 ramified into 5 points.
Proof. For a point c = (c1, c2, c3) ∈ C3, by substituting the Laurent’s solution x(t;m2) of (3.7)
inside equations Fi = ci, i = 1, 2, 3, we find the independent algebraic expressions of t, namely
the three algebraic relations between the parameters a, c, d and e below
c1 = ce2, c2 = a2 − 4c− 48d, c3 = −12a2d+ 48cd+ 16e.
For c ∈ Ω, c1 ̸= 0 therefore the parameters c, d and e are not zero. The first two equations are
linear in the parameters c and d, and can be solved linearly in these parameters as a function
of the constants of motion thus giving
c =
c1
e2
and d =
1
48
(
a2 − c2 −
4c1
e2
)
.
The third equation then reduces to the following equation of an affine curve in C2.
Γ(2)
c : e4a4 −
(
8c1 + c2e
2
)
a2e2 − 64e5 + 4e2c1c2 + 4c3e
4 + 16c21 = 0. (3.14)
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 13
The affine curve Γ
(2)
c is smooth for c ∈ Ω. In fact, let
f(a, e) = e4a4 −
(
8c1 + c2e
2
)
a2e2 − 64e5 + 4e2c1c2 + 4c3e
4 + 16c21,
we have
∂f
∂a
(a, e) = 4a3e4 + 2a
(
−8c1e
2 − c2e
4
)
,
∂f
∂e
(a, e) = 4a4e3 +
(
−16c1e− 4c2e
3
)
a2 − 320e4 + 8c1c2e+ 16c3e
3.
So, a point (a, e) is singular for the affine curve Γ
(2)
c if f(a, e) = ∂f
∂a (a, e) =
∂f
∂e (a, e) = 0 as e ̸= 0
so either a = 0, or a2 = 8c1+c2e2
2e2
.
� If a2 = 8c1+c2e2
2e2
, after substitution in the equations f(a, e) = 0 and ∂f
∂e (a, e) = 0, we then
obtain
−1
4
e4
(
c22 + 256e− 16c3
)
= 0 and − e3
(
c22 + 320e− 16c3
)
= 0.
As e ̸= 0, we have c22 − 16c3 = −256e = −320e, which implies that e = 0, which is absurd!
� If a = 0, this leads to the following system:
−64e5 + 4c1c2e
2 + 4c3e
4 + 16c21 = 0,
−320e4 + 8c1c2e+ 16c3e
3 = 0.
The resultant of these two polynomials in e composing the system is given up to a constant
by the following polynomial in terms of c1, c2 and c3:
c1
(
819200000c21 + 512000c23c2c1 − 57600c3c
3
2c1 − 32c43c
2
2 + c33c
4
2 + 256c53 + 1728c52c1
)
.
This expression is not zero for c ∈ Ω. We deduce that Γ
(2)
c is a smooth affine curve for c ∈ Ω.
The equation (3.14) of the affine curve Γ
(2)
c can be written as(
a2 − 4c1
e2
)(
a2 − 4c1
e2
− c2
)
= 64e− 4c3.
We deduce that Γ
(2)
c is a double cover of the rational affine curve
ε(2)c : u(u− c2)− 64e+ 4c3 = 0,
the map which links the two curves is explicitly given by
ψ : Γ(2)
c −→ ε(2)c , (a, e) 7−→ (u, e) =
(
a2 − 4c1
e2
, e
)
.
ε
(2)
c being irreducible curve of genus 0, a parametrization of ε
(2)
c is given by
ε(2)c =
{
(u, e) =
(
t,
1
64
(
t2 − c2t+ 4c3
))
, t ∈ C
}
.
If ψ(a, e) = (u, e), then we have
a = ±
√√√√ t5 − 2c2t4 +
(
8c3 + c22
)
t3 − 8c2c3t2 + 16c23t+ 16384c1(
t2 − c2t+ 4c3
)2 , e =
1
64
(
t2 − c2t+ 4c3
)
.
14 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
The branch points of the cover ψ : Γ
(2)
c −→ ε
(2)
c are the points (u, e) of the curve ε
(2)
c for which
a = 0, i.e., where u is a root of the polynomial of degree 5 given by
P (t) = t5 − 2c2t
4 +
(
8c3 + c22
)
t3 − 8c2c3t
2 + 16c23t+ 16384c1,
we verify that the discriminant of P (t) is given, up to a constant, by
c21
(
819200000c21 + 512000c1c2c
2
3 − 57600c1c
3
2c3 + 1728c1c
5
2 + 256c53 − 32c22c
4
3 + c42c
3
3
)
,
which for c ∈ Ω is not zero. Therefore, these five branch points are distinct. Thus, the map ψ
admits five ramification points on Γ
(2)
c .
The curve Γ
(2)
c can be completed into a compact Riemann surface, denoted Γ
(2)
c by adding
to it five points to infinity denoted by ∞, ∞ϵ1ϵ2 where ϵ21 = ϵ22 = 1 and δ =
√
c22 − 16c3.
Neighborhoods of these points are described according to a local parameter ς by
∞ϵ1ϵ2 : a = ς−1, e = 2ϵ1
√
c1ς
(
1 +
1
4
(c2 + ϵ2δ)ς
2 +O
(
ς4
))
, (3.15)
∞ : a = ς−1, e =
1
64
(
ς−4 − c2ς
−2 + 4c3 +O
(
ς6
))
. (3.16)
When the application ψ is extended in application ψ : Γ
(2)
c −→ ε
(2)
c , there is another branching
point. Indeed, if we write t = 1
ς2
depending on a local parameter ς, we obtain the point at infinity
∞ ∈ Γ
(2)
c \Γ(2)
c given by
(a, e) =
(
ς−1,
1
64
(
ς−4 − c2ς
−2 + 4c3 +O
(
ς6
)))
.
There is no other branching point. Indeed, noting t = t1 + ς2 and t = t2 + ς2 local parame-
terizations in the neighborhood respectively of t1 and t2 the roots of t2 − c2t + 4c3, it follows
that
a = ±128
ς
√
c1
c
2
2
− 16c3 +O(1),
then respectively
e =
1
64
c2(c2 + δ) +O
(
ς2
)
and e =
1
64
c2(c2 − δ) +O
(
ς2
)
,
which shows that above from the points t1 and t2, the map ψ is not ramified. We then conclude
that the application ψ : Γ
(2)
c −→ ε
(2)
c is a double covering of P1 branched into 5 points. We
deduce that the genus of Γ
(2)
c is equal to 2 according to the Riemann–Hurwitz formula. ■
Remark 3.7. The above propositions only compute an affine part of the divisor D, since the
lower balances depending on 3 free parameters are not used. The completed non singular curves
of respective genus 3, 4 and 2 mentioned in these propositions correspond to blowing up the
singular points of the three irreducible components of D.
3.3 Abelian surface
According to [7], in order to embed the three Riemann surfaces Γ
(0)
c , Γ
(1)
c and Γ
(2)
c into some pro-
jective space, one of the key underlying principles used the Kodaira embedding theorem, which
states that a smooth complex manifold can be smoothly embedded into projective space PN (C)
with the set of functions having a pole of order k along positive divisor on the manifold, pro-
vided k is large enough; fortunately, for abelian surfaces, k need not be larger than three
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 15
according to Lefschetz theorem. These functions are easily constructed from the three Laurent
solutions by looking for polynomials in the phase variables which in the expansions have at most
a k-fold pole. The nature of the expansions and some algebraic properties of abelian varieties
provide a recipe for when to terminate our search for such functions, thus making the procedure
implementable. Precisely, we wish to find a set of polynomial functions {z0, . . . , zN}, of increas-
ing degree in the original variables x0, . . . , y2 having the property that the embedding D of Γ
(i)
c ,
i = 0, 1, 2, into PN (C) via those functions satisfies the relation: g(D) = N +2 where g(D) is the
arithmetic genus of D. If the a
(2)
4 Toda lattice is an irreducible algebraic complete integrable
system, then as we have seen above a divisor D can be added to a Zariski open subset of H,
having the effect of compacting all fibers Fc, where c ∈ Ω. The divisor that is added to Fc will
be denoted by Dc and the resulting torus by T2
c . The vector fields V1 and V2 extend to linear
(hence holomorphic) vector fields on this partial compactification of H, hence we may consider
the integral curves of V1, starting from any component D(i)
c . Since the third power of an ample
divisor on an abelian variety is very ample, we look for all polynomials which have a simple pole
at most when any of the three principal balances are substituted in them. Precisely, we look for
a maximal independent set of functions which are independent when restricted to Fc. By direct
computation, we obtain twenty-five (25) weight homogeneous polynomials of weight at most 13.
These twenty-five (25) functions are defined by
z0 = 1, z11 = x1x2(y0 − 2y2),
z1 = y0, z12 = −x1x2z4,
z2 = y2, z13 = x0x1
(
x1 − y22
)
,
z3 = y22 − x1 − 4x2, z14 = x0x1x2,
z4 = y0y2 − 2x1, z15 = x0x1x2y2,
z5 = y2(y0y2 − 2x1)− 4x2y0, z16 = x0x1(−y0z4 + 4x0y2),
z6 = 2y2
(
y22 − 4x2
)
+ x1(y0 − 4y2), z17 = x1x2((y0 − 2y2)z4 + 8x2y0),
z7 = x0x1, z18 = x0x1x2
(
4x2 − y22
)
,
z8 = x1x2, z19 = x0x1x2
(
4x0 − y20
)
,
z9 = −y0y2z3 + 2x1
(
y22 − x1
)
, z20 = −x0x1x2(−y0z4 + 4x0y2),
z10 = x0x1y2, z21 = −x0x21x2(y0 − 2y2),
z22 = −x0x1x2
(
y0y2(y0y2 − 2x1) + 4x0
(
4x2 − y22
)
− 4x2y
2
0
)
,
z23 = x0x1(4y2x0 + y0(2x1 − y0y2))
(
4x0y
2
2 − y20y
2
2 + 4x1y0y2 − 4x21
)
,
z24 = x2x
3
1x
2
0. (3.17)
For c = (c1, c2, c3) ∈ Ω, we consider the regular map
φc : Fc ⊂ H −→ P24, (x0, . . . , y2) 7−→ (1, z1, . . . , z24), (3.18)
where the functions zi are given by (3.17); this map is embedding of Fc in the projective
space P24(C).
Notice that in this section, the strategy is to embed Fc ∪ (D {singular points}) −→ P24, then
show that the two independent vector fields V1, V2 extend holomorphically to the closure of this
embedding in P24, therefore proving the algebraic complete integrability of the a
(2)
4 Toda lattice,
by the complex Liouville theorem.
16 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
3.3.1 Embedding in the projective space P24
and singularities of the divisor at infinity
In order to show that the a
(2)
4 Toda lattice is algebraic completely integrable, we show that, for
c ∈ Ω, the fiber
Fc := F−1(c) =
3⋂
i=1
{m ∈ H | Fi(m) = ci}
is an affine part of abelian surface, on which the vector fields V1 and V2 restrict to linear vector
fields. To do this, we must check that Fc: satisfies the conditions of the complex Liouville
theorem (Theorem (2.7)). Using (3.18) and let t → 0, we see that the coefficients of t−1 of
series zi(t;m0) define an application
φ(0)
c : Γ(0)
c −→ P24
given by
φ(0)
c : (a, e) 7→ (0 : −2 : 0 : 0 : 2a : 8d− 2a2 : 0 : 0 : e : −2a(−4d+ a2) : −ae : 0 :
− a2e : 0 : de : −ade : −4e(e+ 3ac) : −4ade : de(4d− a2) : 12cde :
4ed(e+ 3ac) : 2de2 : 4de(3a2c+ ae− 12cd) :
− 16ae(3ac+ 2e)(3ac+ e) : e3d), (3.19)
which is, for c ∈ Ω, an embedding of the affine curve Γc. Similarly, the series zi(t;m1)
and zi(t;m2) define two embedding φ
(1)
c and φ
(2)
c of the affine curves Γ
(1)
c and Γ
(2)
c respectively
in the projective space P24 given by
φ(1)
c : (a, e) 7→
(
0 : 2 : 1 : −a : 0 : −6c− a2
2
: −6c+
3a2
2
: d : −8d+ 6ac+
a3
2
:
0 : e : 2ad : −8ea :
d
(
a2 + 12c
)
2
: 0 : 0 : e
(
a2 + 12c
)
:
− d
(
−16d+ 12ca+ a3
)
: −ed : ed : 0 : −2aed : de
(
12c+ a2
)
:
− 1
4
e
(
12c+ a2
)(
a4 + 24ca2 − 16e+ 144c2
)
: de2
)
, (3.20)
φ(2)
c : (a, e) 7→
(
0 : 0 : −2 : 0 : −2a : 0 : 48d : 0 : 24ad : 0 : 0 : 4e : 0 : 2ae :
0 : −2ce : 0 : −2a2e : 0 : 0 : −2ce
(
−4c+ a2
)
: 0 : 0 :
− 8ce
(
−4c+ a2
)2
: 0
)
. (3.21)
Remind that in (3.21) c = c1
e2
and d = 1
48
(
a2 − c2 − 4c1
e2
)
, in (3.20) c = 1
24
(
2a2 − c2
)
and e = c1
d2
,
in (3.19) c = 1
12
(
4a2 − 16d− c2
)
and e2 = c1
d2
.
Looking at the first three coordinates, we observe that the image curves by the embedding φ
(i)
c
are distinct. However, they are not complete, so we check if maybe their closures intersect.
Let us determine the singularities of the divisor at infinity. Let us denote by D(0)
c , D(1)
c
and D(2)
c , respectively, the closures of
φ
(0)
c
(
Γ
(0)
c
)
, φ
(1)
c
(
Γ
(1)
c
)
and φ
(2)
c
(
Γ
(2)
c
)
,
and let Dc =
⋃2
i=0D
(i)
c . Let us determine the singularity of the divisor Dc. To do this, let us
substitute the local parametrization ς around each point at infinity in the corresponding φ
(i)
c
embedding and let ς → 0, we find the following leading terms, where ϵ = ϵ1 = ϵ2 = ϵ3 = ±1.
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 17
By substituting (3.8), (3.9) and (3.10) in (3.19) and considering the first two terms, we have
φ(0)
c (∞ϵ) ∼
(
0 : · · · : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : −c3 :
ϵ
√
c1
4
)
,
φ(0)
c (∞ϵ1ϵ2) ∼
(
0 : 0 : 0 : 0 : 2 : 0 : · · · : 0 :
−c2 + ϵ2δ
4
: 0 : · · · : 0 : ϵ1ϵ2
√
c1 : 0 : 4ϵ1ϵ2
√
c1
0 : 0 :
ϵ2
√
c1(ϵ1c2 + ϵ1ϵ2δ)
2
: 0 : 0 : ϵ1ϵ2
√
c1(ϵ1ϵ2c2 + ϵ1δ)
2 : 0
)
,
φ(0)
c (∞3
ϵ3) ∼
(
0 : · · · : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : −c3 :
ϵ3
√
c1
4
)
. (3.22)
Substituting ϵ = ϵ1 = ϵ2 = ϵ3 = ±1 in (3.22), we have
P+ := lim
p→∞+
φ(0)
c (p) = lim
p→∞3
+
φ(0)
c (p) =
(
0 : 0 : 0 : · · · : 0 : 1 : 0 : · · · : 0 : −c3 :
√
c1
4
)
,
P− := lim
p→∞−
φ(0)
c (p) = lim
p→∞3
−
φ(0)
c (p) =
(
0 : 0 : 0 : · · · : 0 : 1 : 0 : · · · : 0 : −c3 : −
√
c1
4
)
,
Q++ := lim
p→∞++
φ(0)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 :
4
√
c1 : 0 : 0 :
√
c1(c2 + δ)
2
: 0 : 0 :
√
c1(c2 + δ)2 : 0
)
,
Q−+ := lim
p→∞−+
φ(0)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 : −
√
c1 :
0 : −4
√
c1 : 0 : 0 :
√
c1(−c2 − δ)
2
: 0 : 0 : −
√
c1(c2 + δ)2 : 0
)
,
Q+− := lim
p→∞+−
φ(0)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 : −
√
c1 :
0 : −4
√
c1 : 0 : 0 :
√
c1(−c2 + δ)
2
: 0 : 0 : −
√
c1(−c2 + δ)2 : 0
)
,
Q−− := lim
p→∞−−
φ(0)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 :
4
√
c1 : 0 : 0 :
√
c1(c2 − δ)
2
: 0 : 0 :
√
c1(−c2 − δ)2 : 0
)
.
The point P+ is the image of the two points at infinity ∞+ and ∞3
+ while P− is the image of
two points at infinity ∞− and ∞3
−. We deduce that φ
(0)
c does not extend into an embedding
of φ
(0)
c . The curve D(0)
c is therefore singular at the points P ϵ.
By substituting (3.11), (3.12) and (3.13) in (3.20) and consider the two first terms, we obtain
φ(1)
c
(
∞1
)
∼ (0 : 0 : 0 : · · · : 0 : −ς : 0 : 0 : 0 : 1 : 0 : · · · : 0),
φ(1)
c
(
∞2
)
∼
(
0 : 0 : 0 : · · · : 0 :
1
16
ς : −1
2
ς : 0 : 0 : 0 : 0 : − c2
64
ς : 0 : 0 : 0 : 1 : 0 : · · · : 0
)
,
φ(1)
c (∞ϵ) ∼
(
0 : 0 : 0 : · · · : 0 : −4ς : 0 : 0 : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : −c3 : ε
√
c1
4
)
.
Letting ς → 0 and ϵ = ±1, we obtain the points in projective space P24
P+ := lim
p→∞+
φ(1)
c (p) =
(
0 : 0 : 0 : · · · : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : −c3 :
√
c1
4
)
,
18 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
P− := lim
p→∞−
φ(1)
c (p) =
(
0 : 0 : 0 : · · · : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : −c3 : −
√
c1
4
)
,
T := lim
p→∞1
φ(1)
c (p) = lim
p→∞2
φ(1)
c (p) = (0 : 0 : 0 : 0 : · · · : 1 : 0 : 0 : 0 : 0 : 0 : 0 : 0).
The points P+ and P− are distinct but T is the image of two points at infinity ∞1 and ∞2. We
then deduce that φ
(1)
c does not extend into an embedding of φ
(1)
c . The curve D(1)
c is therefore
singular at the point T . By substituting (3.15) and (3.16) in (3.21) and considering that the
first three terms and the fact that ϵ21 = ϵ22 = 1, we obtain the following points in the projective
space P24:
φ(2)
c (∞++) ∼
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 : 4
√
c1 :
0 : 0 :
√
c1(c2 + δ)
2
: 0 : 0 :
√
c1(c2 + δ)2 : 0
)
,
φ(2)
c (∞−+) ∼
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 : 0 : −
√
c1 : 0 : −4
√
c1 :
0 : 0 :
√
c1(−c2 − δ)
2
: 0 : 0 : −
√
c1(c2 + δ)2 : 0
)
,
φ(2)
c (∞+−) ∼
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 : 0 : −
√
c1 : 0 : −4
√
c1 :
0 : 0 :
√
c1(−c2 + δ)
2
: 0 : 0 : −
√
c1(−c2 + δ)2 : 0
)
,
φ(2)
c (∞−−) ∼
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 : 4
√
c1 :
0 : 0 :
√
c1(c2 − δ)
2
: 0 : 0 :
√
c1(−c2 − δ)2 : 0
)
,
φ(2)
c (∞) ∼ (0 : 0 : 0 : · · · : 0 : −ς : 0 : 0 : 0 : 1 : 0 : 0 : 0 : 0 : 0 : 0 : 0).
Hence, making ς → 0, we obtain
Q++ := lim
p→∞++
φ(2)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 :
4
√
c1 : 0 : 0 :
√
c1(c2 + δ)
2
: 0 : 0 :
√
c1(c2 + δ)2 : 0
)
,
Q−+ := lim
p→∞−+
φ(2)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 + δ
4
: 0 : 0 : 0 : 0 : 0 : 0 : −
√
c1 :
0 : −4
√
c1 : 0 : 0 :
√
c1(−c2 − δ)
2
: 0 : 0 : −
√
c1(c2 + δ)2 : 0
)
,
Q+− := lim
p→∞+−
φ(2)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 : 0 : −
√
c1 :
0 : −4
√
c1 : 0 : 0 :
√
c1(−c2 + δ)
2
: 0 : 0 : −
√
c1(−c2 + δ)2 : 0
)
,
Q−− := lim
p→∞−−
φ(2)
c (p) =
(
0 : 0 : 0 : 0 : 2 : 0 : 0 : 0 :
−c2 − δ
4
: 0 : 0 : 0 : 0 : 0 : 0 :
√
c1 : 0 :
4
√
c1 : 0 : 0 :
√
c1(c2 − δ)
2
: 0 : 0 :
√
c1(−c2 − δ)2 : 0
)
,
T := lim
p→∞
φ(2)
c (p) = (0 : 0 : 0 : · · · : 0 : 1 : 0 : · · · : 0).
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 19
D2
D1
D0
Figure 1. Curves completing the invariant surfaces Fc of the a
(2)
4 Toda lattice in abelian surfaces,
where Di is the curve D(i)
c .
The points Qϵ1ϵ2 , T are all distinct. We then deduce that φ
(2)
c extends into an embedding
of φ
(2)
c . The curve D(2)
c is therefore not singular. The singularities of the curves can be seen
from Figure 1. Notice that Figure 1 shows the divisor and coincides with that found by M. Adler
and P. van Moerbeke (see [3, Table 2]). Hence the conjecture is verify.
3.3.2 The quadratic differential equations and holomorphic
Now we want to show that the vector field (φc)∗V1 extends into a holomorphic vectors field
on P24. This is done by exhibiting the quadratic differential equations in two of the charts. We
will use the following
Lemma 3.8 ([4]). Let X be a vector field on PN holomorphic in two different maps (Zi ̸= 0)
and (Zj ̸= 0). Then X is holomorphic over PN . That is to say on any map (Zj ̸= 0).
Proposition 3.9. The vector field (φc)∗V1 extends into a holomorphic vectors field on P24.
Proof. It suffices to show that the vector field (φc)∗V1 is holomorphic on two chart of P24. To
do this, let us establish that this vector field can be written as a quadratic vector field in the
chart Z0 ̸= 0 and Z1 ̸= 0. In the chart Z0 ̸= 0, we obtain the following result:
ż1 = −1
2
(c2 − 4(z3 + z4) + z1(4z2 − z1)),
ż2 =
1
4
(3(z4 − z1z2) + 2(z22 − z3)),
ż3 =
1
2
z6 − z2z3,
ż4 =
1
2
(z4(z1 + z2)− c2z2 + 2z6 − z5),
ż5 =
1
2
(z1z5 − c3) + 2(6z7 − z9 − z2z5),
ż6 = −1
2
(z2z5 + z6(z1 − z2) + 3z9) + 2(4z7 + z8)− z23 ,
ż7 = −1
2
(z11 + 2z2z7),
ż8 =
1
2
z1z8 − z10,
ż9 =
1
2
(z2(c2z3z + z9 − 24z7) + z5(z3 − z4))− 4z11 − z3z6,
ż10 =
1
2
(−2z2z10 + z4z8 + 4z14),
ż11 =
1
2
(z13 + 2(z3z7 + 2z14) + z11(z2 − z1)),
20 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
ż12 =
1
2
(2z3z10 − z5z8 − 4z1z14),
ż13 =
1
2
(z13(z2 − z1) + z5z7 − 4z15),
ż14 =
1
2
z1z14,
ż15 =
1
2
(z18 − 2z7z8 + z15(z1 + z2)),
ż16 = ∗,
ż17 = ∆,
ż18 = −1
2
z5z14 + z7z10,
ż19 = 4z7z10 −
1
2
z14(z1 − 4z5),
ż20 =
1
2
(z22 + z2z20 − z4z19 + 4z8(z13 − 2z14)),
ż21 = 2z7(z12 + 2z14)− z8(z13 + 2z14),
ż22 =
1
2
(z1z22 − z7z16)− 2(z10z13 + 6z1c1),
ż23 = ⋆,
ż24 =
1
2
(z24(z1 − 4z2))
with
∗ =
1
4
(z16(z1 − 4z2) + z8(32z7 + 2z8 − c2z4) + 4(2z19 + z5z10)),
⋆ =
1
2
(z23(z1 + z2) + 5z16(8z11 + c3z2)) + 128z14(z13 + 2z14) + c3z8(c3 − 32z7)
+ 64c1z1(2z2 − z1)− 4c3z19,
∆ =
1
2
(4(3z18 + 2z1z15)− z1z17 − z4z13 − z7(8z8 + c3)).
For the second chart Z1 ̸= 0, we put si = zi/z1, for all i = 0, . . . , 24. Then the quadratic
differential equations take the following form:
ṡ0 =
1
2
(s0(c2s0 − 4(s3 + s4)) + 4s2 − 1),
ṡ1 = 0,
ṡ2 =
1
2
(s0(c2s2 − s5 − 4s6) + s2(4s3 + s4)− s4),
ṡ3 =
1
2
(s0(c3s0 − 32s7 + 4s9)− s5 + s6 + 2s2s5),
ṡ4 =
1
2
(
s0(8s8 + 6s9) + s5(6s2 − 1) + s24
)
,
ṡ5 = 4s0(s10 + s11)− s9 + 8s2s7 − s4s5,
ṡ6 =
1
2
s0(16s10 − 24s11 − c2s6) + s2(2c2s3 − 16s7 − s9)− s3(s5 + s6)− 2s8 + s9,
ṡ7 = s0(s13 − 2s14)−
1
2
s11,
ṡ8 =
1
2
(s0(c2s8 − 32s14)) + s10(4s2 − 1)− 6s3s8,
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 21
ṡ9 =
3
4
(s0(c3s4 + 32s14 + 16s13)− c3s2)
+
1
2
(
s2(2(c3s2 + 8s11 − 2s10)− c2s6) + s4(s9 − 4s8) + s25 + 2s26 − 16s11
)
,
ṡ10 = −1
2
s0s16 − 2s14 − s5s8,
ṡ11 = −2s14 + 4s0s15 + s13(s2 − 1)− 1
2
s7(c2 + 6s5),
ṡ12 =
1
8
(c2s4s8 + s16(4s2 + 1))− s8(4s7 + 3s8),
ṡ13 =
1
2
(s4s13 + s17 + s7(3c3s0 − 48s7 − 4s8))− 6s0s17,
ṡ14 = −1
2
(s0s19 − 4s7s8),
ṡ15 =
1
2
(s0s20 + s4s15 − s5s14 + 4s7s10),
ṡ16 = 2(8s18 + 3s19) +
1
2
s16(c2s0 − 4s4) + 2s10(8s8 + c3s0 + 32s7)
− s5(c2s8 + 4s12)− 32s0s21,
ṡ17 = 2s14(s5 − 2s6) + s11(4s8 − c2s3) +
1
2
s7(c2(2s5 − c2s2) + c3 + 8s10)− 3s5s13
− 2s0s20 +
1
2
s4s17,
ṡ18 =
1
2
(s3s19 + s21) + 8c1s
2
0 + 2s7s12 + s8(2s14 − s13),
ṡ19 =
1
2
c2s0s19 − 2s0s22 + 24s8s14 + 2s21,
ṡ20 =
1
2
(s22 + s4s20 − c2s2s19) + s6s19 − s7s16 − 8s8s15,
ṡ21 = 4c1s0 + 2s10s13 − s7s16,
ṡ22 =
1
2
(s0(c3s19 − 16s24)) + 8
(
s10s15 − 8s214
)
+ 4c1,
ṡ23 =
1
2
(s23(s4 + c2s0)− c3(s8(3c3 + 40s10)− 2s3s16 − 4s19(4s2 + 3)))− 96s24(1 + 2s2)
+ 8s16(4s13 + 10s14 − c2s7)− 192c1s4 + 160s11s19 − 96s24,
ṡ24 =
1
2
(s10s20 − s12s19 − 2s4s24).
This shows that the vector field (φc)∗V1 extends into a linear vector field V1 on P24. ■
Thus, the item four of the complex Liouville theorem (Theorem (2.7)) is satisfied. We now
show that the integral curves of V1 that start at the three singular points go into the affine
immediately
Proposition 3.10. The flow Φt of the vector field V1 on P24 coming from the points of Dc is
sent into the affine part φc(Fc).
Proof. Note that this is automatic for the points of φc
(
Γ
(0)
c
)
, φc
(
Γ
(1)
c
)
and φc
(
Γ
(2)
c
)
. We then
just need to check for the singular points Qε1ε2 , Pε and T in P24.
Consider the 4 points Qε1ε2 intersections of D(0)
c and D(2)
c . From (3.9) and (3.19) follow
that the leading coefficient of z4
(
t;Γ
(0)
c
)
has a pole for ς = 0, that is maximal with the leading
coefficient of z9
(
t;Γ
(0)
c
)
and thus the function z4 defines a map at these points. Let us show then
that limp→∞ϵ1ϵ2
1
z4
φ
(0)
c ̸= 0.
22 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
Using (3.5), we get
z4(t;m0) =
2a
t
− 4d+ (−e+ 2ad− 2ac)t+
1
6
(
ae− 4
(
2d2 + da2
)
+ 4cd
)
t2 +O
(
t3
)
.
The first terms of the inverse of this series are then given by
1
z4(t;m0)
=
t
2a
+
d
a2
t2 +O
(
t3
)
. (3.23)
Substituting e = −4a4−32a2d−a2c2+64d2+4dc2+c3
8a , c = a2
3 − 4d
3 − c2
12 in the second term and
rewriting the coefficients according to the local parameter ς around ∞ϵ1ϵ2 using (3.9), we obtain
lim
ς→0
1
z4
(t, ς) =
1
4
t2 +O
(
t3
)
̸= 0,
which prove that the integrals curves of V1 which start from the points Qϵ1ϵ2 are immediately
sent in the affine part φc(Fc).
For the point Pϵ, intersection points of D(0)
c and D(1)
c , the residue having the largest pole
among the residues of z0
(
t;Γ
(1)
c
)
, . . . , z24
(
t;Γ
(1)
c
)
is z9
(
t;Γ
(1)
c
)
, the function z15 defines a local
chart around this point. Consider the balances x(t;m1) 3.6 and rewriting a, b, c, d of 1/z9
(
t;Γ
(1)
c
)
depending on the local parameter ς around ∞ϵ using c = 1
12a
2 − 1
24c2, e =
c1
d2
and (3.11), we
obtain
lim
ς→0
1
z9
(t, ς) =
1
12
t4 +O
(
t6
)
̸= 0.
Consider the balances x(t;m0) (3.5) and the local parametrization (3.8), we also obtain
lim
ς→0
1
z8
(t, ς) =
1
12
t4 +O
(
t6
)
̸= 0.
Thus, the different limits found are different from zero, which also shows that the integral curve
of V1 which starts from the points Pϵ are immediately sent in the affine part φc(Fc).
For the point T , intersection point of D(2)
c and D(1)
c , the only non-zero coordinate corresponds
to the function z17
(
t;Γ
(1)
c
)
, the function z17 defines a local map around this point. Consider the
balances x(t;m1) (3.6) and x(t;m2) (3.7) by writing a, c, d, e of 1/z17
(
t;Γ
(1)
c
)
depending on the
local parameter ς around ∞ϵ and using c = 1
12a
2 − 1
24c2, e =
c1
d2
and (3.11), we get
lim
ς→0
1
z17
(t, ς) =
1
2304
t7 +O
(
t8
)
̸= 0.
Considering the balances x(t;m2) (3.7) and local parametrization (3.16), we also get
lim
ς→0
1
z17
(t, ς) = − 1
1152
t7 +O
(
t8
)
= −2×
(
1
2304
t7
)
+O
(
t8
)
̸= 0.
Thus, the different limits found are different from zero, which also shows that the integral curve
of V1 which starts from the points T are immediately sent in the affine part φc(Fc).
Thus, the flow of the vector field V1 starting from each of the points of D(0)
c ∪ D(1)
c ∪ D(2)
c
goes into the affine part of φc(Fc).
In order to complete the proof of algebraic integrability, it is necessary to show that there
exist no other divisors passing through the points Qϵ1ϵ2 , Pϵ and T .
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 23
For the four intersections points Qϵ1ϵ2 of D(0)
c and D(2)
c , by rewriting the coefficients of the
right-hand side of (3.23) with respect to the local parameter ς in the neighborhood of the points
∞ϵ1ϵ2 ∈ Γ
(0)
c , we find
1
z4
(
t; Γ
(0)
c
) =
1
4
(
2ςt+ t2
)
+O
(
t3, ςt2
)
,
which shows that the multiplicity of 1
z4
at each of these points is equal to 2, which coincides with
the sum of the orders of zero of 1
z4
on each of the divisors so there are no other divisors passing
through the points Qϵ1ϵ2 .
For the points Pϵ and T of the divisor Dc, obtained from the points at infinity∞ϵ, ∞1 and∞2,
let us check that the degree of Dc which is 3 is indeed equal to the degree of φc(Fc) \ φc(Fc)
at these points. As the vector field V1 is only tangent to one of the branches (transverse to
the other) of D1
c passing through these two points, we do the expansion along the non-tangent
branch. As the function z17 defines a map at point T , we just need to substitute (3.11) into its
inverse series,
1
z17
(
t;Γ
(1)
c
) = − t
d
(
−16d+ 12ac+ a3
) +O
(
t2
)
= − 2t
d
(
−32d− ac2 + 4a3
) +O
(
t2
)
,
which leads to
1
z17
(
t;Γ
(1)
c
) = − 1
4
√
c1
ςt+O
(
t3
)
,
thus showing that there are no other divisors passing through T .
For points Pϵ, we do this by calculating the first terms of the series 1/z8
(
t;Γ
(0)
c
)
using
Laurent’s solution x(t;m0) then we express the free parameters according to the local parameter ς
in a neighborhood of (3.8). The resulting series in ς and t should start with monomials of degree 3
because the point Pϵ has multiplicity 2 and 1 on the divisors D(0)
c and D(1)
c , respectively, and
the function z8 has a simple pole on each of these divisors. We have
1
z8
(
t;Γ
(0)
c
) =
1
e
t− at2
e
+O
(
t3
)
,
which leads to
1
z8
(
t;Γ
(0)
c
) = 2ς2t2 +O
(
ς3, ςt3
)
,
thus showing that there are no other passing divisors by Pϵ. Furthermore this also shows that Pϵ
is an ordinary double point for the divisor D(0)
c . So, there are no other divisors in φc(Fc) \ φc(Fc)
besides the divisors D(0)
c , D(1)
c and D(2)
c already found. ■
The Liouville complex theorem conditions being satisfied, it follows that for c ∈ Ω, the pro-
jective variety φc(Fc) = φc(Fc) ∪ Dc is an abelian surface and the restrictions of vector fields V1
and V2 to these abelian surfaces are linear. Since φc(Fc) contains a smooth curve of genus 2, it
is the Jacobian of this curve. We have therefore proved the following theorem.
Theorem 3.11. Let (H, {·, ·},F) be an integrable system describing the a
(2)
4 Toda lattice, where
F = (F1, F2, F3) and {·, ·} are given, respectively, by (3.3) and (3.2) with commuting vector
fields (3.1).
(i) (H, {·, ·},F) is a weight homogeneous algebraical completely integrable system.
24 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
(ii) For c ∈ Ω, the fiber Fc of its momentum map is completed in an abelian surface T2
c
(
the
Jacobian of the hyperelliptic curve (of genus two) Γ
(2)
c
)
by the addition of a singular divi-
sor Dc composed of three irreducible components: D(0)
c defined by
Γ(0)
c : 16d2a8 −
(
256d3 + 8d2c2
)
a6 +
(
1536d2 + 96dc2 + 8c3 + c22
)
d2a4
−
((
8
(
8c3 + 48dc2 + c22 + 512d2
)
d+ 2c2c3
)
d2 + 64c1
)
a2
+
(
8d
(
c2c3 + 16dc3 + 64d2c2 + 512d3 + 2dc22
)
+ c23
)
d2 = 0,
and D(1)
c defined by
Γ(1)
c : 256ad3 −
((
4a2 − c2
)2 − 16c3
)
d2 + 64c1 = 0,
two singular curves of respective genus 3 and 4 and one smooth curve and D(2)
c defined by
Γ(2)
c : e4a4 −
(
8c1 + c2e
2
)
a2e2 − 64e5 + 4e2c1c2 + 4c3e
4 + 16c21 = 0
of genus 2 and isomorphic to Γ
(2)
c . The curves intercept each other as indicated in Figure 1.
4 Geometry of the a
(2)
4 Toda lattice:
holomorphic differentials forms
In this section, we show following an idea by Luc Haine (see [9]) how the holomorphic differentials
are used to determine the tangency locus of the vector field V1 on the divisor Dc, we compute the
holomorphic differentials ω1 and ω2 on the three irreducible components of Dc that come from
the differentials dt1 and dt2 on the abelian surface T2
c . We know that all irreducible components
have multiplicity 1.
Let us calculate the holomorphic differential forms ω1 and ω2 on the divisor Dc which come
from the differentials dt1 and dt2 on the abelian surface T2
c . Let y0 := z1 and y := z4 be
restricted to D(0)
c , the first coefficients of their Laurent series are given by
y
(0)
0 = −2, y
(1)
0 = 2c, y(0) = 2a, y(1) = −4d,
and using (3.5) and (3.4), we have
V2
[
1
z1
]
|D(0)
c
= −2a2 + 8d, V2
[
z4
z1
]
|D(0)
c
= −4ea− 24cd.
It then follows that
δ =
1(
y
(0)
0
)2
∣∣∣∣∣∣ y
(0)
0 V2
[
1
y0
]
|D(0)
c
y
(0)
0 y(1) − y(0)y
(1)
0 V2
[
z4
y0
]
|D(0)
c
∣∣∣∣∣∣ = 1
4
∣∣∣∣ −2 −2a2 + 8d
8d− 4ac −4ea− 24cd
∣∣∣∣
= 2ea+ 12cd+ 4a2d− 2a3c− 16d2 + 8cad.
The holomorphic differential forms dt1 and dt2 restricted to D(0)
c are given by
ω1 =
1
δy
(0)
0
d
(
y(0)
y
(0)
0
)
= −da
δ
and ω2 = −1
δ
V2
[
1
y0
]
|D(0)
c
d
(
y(0)
y
(0)
0
)
= −2a2 − 8d
δ
da.
To determined differential forms on the divisor D(1)
c , consider the functions y0 := z1 and
y := z3, restricted to D(1)
c , the first coefficients of their Laurent series are given by
y
(0)
0 = 2, y
(1)
0 = a, y(0) = −a, y(1) = −3c+
a2
4
,
Algebraic Complete Integrability of the a
(2)
4 Toda Lattice 25
and using (3.6) and (3.4), we have
V2
[
1
z1
]
|D(1)
c
= −6c− a2
2
, V2
[
z3
z1
]
|D(1)
c
= −2e+ 4ad+ 18c2 − 3a2c− 3a4
8
,
then
δ =
1(
y
(1)
0
)2
∣∣∣∣∣∣ y
(0)
0 V2
[
1
y0
]
|D(1)
c
y
(0)
0 y(1) − y(0)y
(1)
0 V2
[
z3
y0
]
|D(1)
c
∣∣∣∣∣∣
=
1
4
∣∣∣∣∣ 2 −6c− a2
2
−6c+ 3a2
2 −2e+ 4ad+ 18c2 − 3a2c− 3a4
8
∣∣∣∣∣ = 2ad− e.
The holomorphic differential forms dt1 and dt2 restricted to D(1)
c are given by
ω1 =
1
δy
(0)
0
d
(
y(0)
y
(0)
0
)
= −da
4δ
and ω2 = −1
δ
V2
[
1
y0
]
|D(1)
c
d
(
y(0)
y
(0)
0
)
= −a
2 + 12c
4δ
da.
For calculating differential forms on the divisor D(2)
c , consider the functions y0 := z2 and
y := z4 restricted to D(2)
c , the first coefficients of their Laurent series are given by
y
(0)
0 = −2, y
(1)
0 = 2d, y(0) = −2a, y(1) = −4c,
and using (3.7) and (3.4), we have
V2
[
1
z2
]
|D(2)
c
= 2c− a2
2
, V2
[
z4
z2
]
|D(2)
c
= 16e− 96cd.
It then follows that
δ =
1(
y
(1)
0
)2
∣∣∣∣∣∣ y
(0)
0 V2
[
1
y0
]
|D(2)
c
y
(0)
0 y(1) − y(0)y
(1)
0 V2
[
z3
y0
]
|D(2)
c
∣∣∣∣∣∣ = 1
4
∣∣∣∣ −2 2c− a2
2
8c+ 4ad 16e− 96cd
∣∣∣∣
= 48cd− 8e+ ca2 +
a3d
2
− 4c2 − 2acd.
The holomorphic differential forms dt1 and dt2 restricted to D(2)
c are given by
ω1 =
1
δy
(0)
0
d
(
y(0)
y
(0)
0
)
= −da
2δ
and ω2 = −1
δ
V2
[
1
y0
]
|D(2)
c
d
(
y(0)
y
(0)
0
)
= −4c− a2
2δ
da.
The zero of differentials forms ω1 and ω2 gives the tangency points of fields vector V1 and V2
respectively.
Remark 4.1. In this paper, we use the Maple 13 application to develop and implement algo-
rithms for determining Laurent series, various curves. We also use this application to determine
the Painlevé divisors, the zi functions, embedding to the projective space and give the different
quadratic differential equations of the two chart Z0 and Z1.
Acknowledgements
We would like to extend our sincere gratitude to Professor Pol Vanhaecke at University of
Poitiers for his particular contributions in providing clarifications and guidance on our research
theme, for the enriching exchanges and thoughtful advice he generously offered us throughout
this project. We cannot end our acknowledgements without thanking all the referees of this
paper. We wish to express our thanks to the referees for their valuable helpful comments and
suggestions.
26 B.L. Lietap Ndi, D. Dehainsala and J. Dongho
References
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https://doi.org/10.1016/0001-8708(80)90008-0
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https://doi.org/10.1007/BF01239513
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https://doi.org/10.1143/JPSJ.20.2095A
https://doi.org/10.1143/PTPS.36.113
1 Introduction
2 Preliminaries
3 Algebraic integrability of the a_4^(2) Toda lattice
3.1 Liouville integrability of the a_4^(2) Toda lattice
3.2 Algebraic integrability of a_4^(2) Toda lattice
3.2.1 Laurent solutions
3.2.2 Painlevé divisors of a_4^(2) Toda lattice
3.3 Abelian surface
3.3.1 Embedding in the projective space P^24 and singularities of the divisor at infinity
3.3.2 The quadratic differential equations and holomorphic
4 Geometry of the a_4^(2) Toda lattice: holomorphic differentials forms
References
|
| id | nasplib_isofts_kiev_ua-123456789-212608 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:36:29Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lietap Ndi, Bruce Lionnel Dehainsala, Djagwa Dongho, Joseph 2026-02-09T08:05:46Z 2024 Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice. Bruce Lionnel Lietap Ndi, Djagwa Dehainsala and Joseph Dongho. SIGMA 20 (2024), 087, 26 pages 1815-0659 2020 Mathematics Subject Classification: 34G20; 34M55; 37J35 arXiv:2404.13688 https://nasplib.isofts.kiev.ua/handle/123456789/212608 https://doi.org/10.3842/SIGMA.2024.087 This work aims to investigate the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra ⁽²⁾₄. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity. We would like to extend our sincere gratitude to Professor Pol Vanhaecke at the University of Poitiers for his particular contributions, including clarifications and guidance on our research theme, and for the enriching exchanges and thoughtful advice he generously offered us throughout this project. We cannot end our acknowledgements without thanking all the referees of this paper. We wish to express our thanks to the referees for their valuable and helpful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice Article published earlier |
| spellingShingle | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice Lietap Ndi, Bruce Lionnel Dehainsala, Djagwa Dongho, Joseph |
| title | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice |
| title_full | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice |
| title_fullStr | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice |
| title_full_unstemmed | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice |
| title_short | Algebraic Complete Integrability of the ⁽²⁾₄ Toda Lattice |
| title_sort | algebraic complete integrability of the ⁽²⁾₄ toda lattice |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212608 |
| work_keys_str_mv | AT lietapndibrucelionnel algebraiccompleteintegrabilityofthe24todalattice AT dehainsaladjagwa algebraiccompleteintegrabilityofthe24todalattice AT donghojoseph algebraiccompleteintegrabilityofthe24todalattice |