## INTEGRABILITY OF HAMILTONIAN SYSTEMS AND TRANSSERIES EXPANSIONS

WERNER BALSER AND MASAFUMI YOSHINO

Abstract. This paper studies analytic Liouville-non-integrable and*C*^{∞}-Liouville-
integrable Hamiltonian systems with two degrees of freedom. We will show that
considerably general Hamiltonians than the one studied in [1] have the property.

We also show that a certain monodromy property of an ordinary diﬀerential equa- tion obtained as a subsystem of a given Hamiltonian and the transseries expansion of a ﬁrst integral play an important role in the analysis. In the former half we will show that the analytic Liouville-non-integrability holds for a rather wider class of Hamiltonians than in [1] under a certain monodromy condition. For these analytic non integrable Hamiltonians we cannot construct nonanalytic ﬁrst integrals con- cretely as in [1]. In the latter half, we show the nonanalytic integrability from the viewpoint of a transseries expansion of a ﬁrst integral. We will construct a ﬁrst inte- gral in transseries formally under general situation. Then we discuss convergence or existence of the ﬁrst integral which has a given formal transseries as an asymptotic expansion.

1. Introduction

A Hamiltonian system in*n* degrees of freedom is said to be*C*^{∞}-Liouville-integrable
if there are *n* smooth ﬁrst integrals in involution which are functionally independent
on an open dense set. If the ﬁrst integrals are analytic, then we say that it is analytic-
Liouville-integrable. In the paper [1], Gorni and Zampieri showed the existence of
a Hamiltonian system with two degrees of freedom which is not analytic-Liouville-
integrable, while it is *C*^{∞}-Liouville-integrable. The geometrical motivation to study
such an example comes from the integrability of a geodesic ﬂows and the Taˇimanov’s
problem. (cf. [1]). We note that the proof of analytic-nonintegrability relies on the
power series expansion of a ﬁrst integral, and the *C*^{∞}- integrability was proved by
constructing concretely a smooth ﬁrst integral. (cf. Remark after Corollary 2.2.) In
this paper, we are interested in the analytical structures which yield nonintegrability
in the framework of rather general Hamiltonians than those in previous works. In
fact, we will show that the monodromy property of a certain subsystem of a given
Hamiltonian plays an important role.

*Date*: July 26, 2009.

2000*Mathematics Subject Classiﬁcation.* Primary 35C10; Secondary 40G10, 35Q15 .
*Key words and phrases.* nonintegrability, Hamiltonian systems, transseries, summability.

Institut f¨ur Angewandte Analysis, Universit¨at Ulm, D–89069 Ulm, Germany

Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 Japan.

1

For these analytic-nonintegrable Hamiltonians, we cannot construct a nonanalytic ﬁrst integral concretely, and instead we make use of a transseries in order to con- struct such an integral. An integral in a transseries expansion is constructed via the Lagrange-Charpit system of a certain vector ﬁeld obtained by restricting a given Hamiltonian vector ﬁeld to an invariant manifold. The construction of a ﬁrst integral as a formal transseries is elementary, while the convergence part is complicated due to the degeneracy of a given Hamiltonian. We will study the convergence from two dif- ferent points of view, transformation of transseries and Borel summability argument for transseries.

This paper is organized as follows. In *§*2 we study the necessity for the analytic-
Liouville-nonintegrability under the monodromy condition. In *§*3 we give the proof
of Theorem 2.1. In *§*4 we give the transseries expansion of the integrals in a formal
sense. The convergence in a curved region and transformation of transseries is dis-
cussed in *§*5. In *§*6 we study the asymptotic property of transseries in terms of Borel
summability method.

2. Analytic nonintegrability

Let*σ≥*1 be an integer and let*r*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) be an analytic function of (*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*∈*
R^{4} in some neighborhood of the origin 0*∈*R^{4} such that

*r≡r*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) =*q*_{1}^{2}^{σ}+*a*(*q*_{1}^{2}^{σ})*q*_{2}^{2}+ ˜*r*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*q*_{2}^{3}*,*
(2.1)

where ˜*r*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) is analytic at the origin and *a*(*t*) (*t* = *q*^{2}_{1}^{σ}) is a polynomial of *t*
such that *a*(0) *>* 0. We are interested in the following analytic Hamiltonian in R^{4}
with two degrees of freedom

*H* =*−q*_{2}*p*_{2}*∂**q*_{1}*r*+

*r*^{2}+*q*_{2}*∂**q*_{2}*r*
*p*_{1}*,*
(2.2)

where *∂**q*_{1} = _{∂q}^{∂}

1 and *∂**q*_{2} = _{∂q}^{∂}

2. The associated Hamiltonian system is given by

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

˙

*q*_{1} =*∂H/*(*∂p*_{1}) =*r*^{2}+*q*_{2}*∂**q*_{2}*r*+ (2*r∂**p*_{1}*r*+*q*_{2}*∂**q*_{2}*∂**p*_{1}*r*)*p*_{1}*−q*_{2}*p*_{2}*∂**p*_{1}*∂**q*_{1}*r,*

˙

*q*_{2} =*∂H/*(*∂p*_{2}) =*−q*_{2}*∂*_{q}_{1}*r−q*_{2}*p*_{2}*∂*_{q}_{1}*∂*_{p}_{2}*r*+*p*_{1}(2*r∂*_{p}_{2}*r*+*q*_{2}*∂*_{q}_{2}*∂*_{p}_{2}*r*)*,*

˙

*p*_{1} =*−∂H/*(*∂q*_{1}) =*q*_{2}*p*_{2}*∂*_{q}^{2}

1*r−*(2*r∂*_{q}_{1}*r*+*q*_{2}*∂*_{q}_{1}*∂*_{q}_{2}*r*)*p*_{1}*,*

˙

*p*_{2} =*−∂H/*(*∂q*_{2}) =*p*_{2}*∂**q*_{1}*r*+*q*_{2}*p*_{2}*∂**q*_{1}*∂**q*_{2}*r−*

2*r∂**q*_{2}*r*+*∂**q*_{2}*r*+*q*_{2}*∂*_{q}^{2}_{2}*r*
*p*_{1}*.*
(2.3)

We need a deﬁnition in order to state our theorem.

Definition 2.1**.** *We say that a polynomial* *a*(*t*) *satisﬁes the monodromy condition*
*(M) if the following equation has no polynomial solution* *U*(*t*)

2*σt*^{2}*U*^{}*−*4*σU*+ (1*−*6*σ*)*tU* = (*t*+ 1)*a*(*t*)*.*

(2.4)

Then we have

Theorem 2.1**.** *Assume that* *a*(*t*) *satisfy (M). Then the Hamiltonian system (2.3) is*
*not analytic-Liouville-integrable in any neighborhood of the origin. More precisely, for*
*any analytic ﬁrst integral* *u*=*u*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) *of (2.3) in* R^{4}*, there exists a function* *φ*
*of one-variable, being analytic at* 0*∈*R *such that* *u*=*φ◦ H.*

If *a*(*t*) *≡* 1, then we can easily see, from the direct computations or by Lemma 2
of [4] that (M) is satisﬁed. Hence we have

Corollary 2.2**.** *Suppose that* *σ* = 1*,* *a*(*t*) *≡* 1 *and* *r*˜ *≡* 0 *in (2.1). Then the*
*Hamiltonian system (2.3) is not analytic-Liouville-integrable in any neighborhood of*
*the origin.*

**Remark.** (a) As to the fundamental properties of (M) we refer [4]. In this paper,
we change the terminology for the sake of simplicity. We remark that (M) is a generic
condition.

(b) Theorem 2.1 is a generalization of [4, Theorem 1], where the function *r*in (2.1)
was supposed to be independent of*p*_{1} and*p*_{2}. Corollary 2.2 was proved in [1]. In this
case, it is not diﬃcult to see that (2.3) in Corollary 2.2 is *C*^{∞}-Liouville-integrable,
because it has a smooth ﬁrst integral

*u*=

*q*_{2}exp

*−*^{1}_{r}

if (*q*_{1}*, q*_{2})= (0*,*0)*,*
0 if (*q*_{1}*, q*_{2}) = (0*,*0)*.*

(2.5)

On the other hand, it is not known whether (2.3) in a general case has a nonanalytic
ﬁrst integral because one cannot construct the ﬁrst integral of (2.3) concretely since
*r* also depends on*p*_{1} and*p*_{2}. In *§*4 we will study the integrability from the viewpoint
of transseries.

3. Proof of theorem

The proof of Theorem 2.1 is done by the argument in [4]. For the sake of complete- ness we give the proof.

*Proof of Theorem 2.1.* By the suitable change of the variable*q*_{2}one may assume that
*a*(0) = 1. Let *u*=*u*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) be any analytic ﬁrst integral of (2.3). We note that
*u* is the ﬁrst integral of the Hamiltonian system (2.3) if and only if*u* is a solution of
the following ﬁrst order equation

*{H, u} ≡*

*q*_{2}*p*_{2}*∂*_{q}^{2}_{1}*r−*(2*r∂**q*_{1}*r*+*q*_{2}*∂**q*_{1}*∂**q*_{2}*r*)*p*_{1} *∂u*

*∂p*_{1}
(3.1)

+

*p*_{2}*∂**q*_{1}*r*+*q*_{2}*p*_{2}*∂**q*_{1}*∂**q*_{2}*r−*

2*r∂**q*_{2}*r*+*∂**q*_{2}*r*+*q*_{2}*∂*_{q}^{2}_{2}*r*

*p*_{1} *∂u*

*∂p*_{2}

+

*r*^{2}+*q*_{2}*∂*_{q}_{2}*r*+ (2*r∂*_{p}_{1}*r*+*q*_{2}*∂*_{q}_{2}*∂*_{p}_{1}*r*)*p*_{1}*−q*_{2}*p*_{2}*∂*_{p}_{1}*∂*_{q}_{1}*r* *∂u*

*∂q*_{1}
+ (*−q*_{2}*∂*_{q}_{1}*r−q*_{2}*p*_{2}*∂*_{q}_{1}*∂*_{p}_{2}*r*+*p*_{1}(2*r∂*_{p}_{2}*r*+*q*_{2}*∂*_{q}_{2}*∂*_{p}_{2}*r*)) *∂u*

*∂q*_{2} = 0*.*

We deﬁne

*v* *≡v*(*q*_{1}*, p*_{1}*, p*_{2}) :=*u*(*q*_{1}*,*0*, p*_{1}*, p*_{2})*.*

(3.2)

By setting *q*_{2} = 0 in (3.1) and noting that *∂*_{q}_{2}*r*(*q*_{1}*,*0)*≡*0 and *r*(*q*_{1}*,*0) =*q*_{1}^{2}^{σ} by (2.1),
we obtain

2*σp*_{2} *∂v*

*∂p*_{2} *−*4*σq*_{1}^{2}^{σ}*p*_{1} *∂v*

*∂p*_{1} +*q*^{2}_{1}^{σ}^{+1} *∂v*

*∂q*_{1} = 0*.*

(3.3)

We expand *v* into the power series of *p*_{2}, *v* = ^{∞}_{j}_{=0}*v**j*(*q*_{1}*, p*_{1})*p*^{j}_{2}. Then we see that
*v*_{j}(*q*_{1}*, p*_{1}) (*j* = 0*,*1*, . . .*) satisfy

2*σjv*_{j} *−*4*σq*^{2}_{1}^{σ}*p*_{1}*∂v**j*

*∂p*_{1} +*q*_{1}^{2}^{σ}^{+1}*∂v**j*

*∂q*_{1} = 0*,* *j* = 0*,*1*,*2*, . . .*
(3.4)

We want to show that *v*_{j} = 0 if *j* = 0, and *v* = *φ*(*p*_{1}*q*_{1}^{4}^{σ}) for some analytic func-
tion *φ*(*t*) of one variable. Indeed, by substituting the expansion of *v*_{j} *v*_{j}(*q*_{1}*, p*_{1}) =

*∞**ν*=0*v*_{j,ν}(*q*_{1})*p*^{ν}_{1} into (3.4) we obtain

2*σjv*_{j,ν}*−*4*σνq*_{1}^{2}^{σ}*v*_{j,ν}+*q*^{2}_{1}^{σ}^{+1}*∂v*_{j,ν}

*∂q*_{1} = 0*,* *j* = 0*,*1*,*2*, . . .*
(3.5)

If we expand *v**j,ν* into the power series of *q*_{1}, then we can easily see that *v**j,ν* *≡*0 for
all*ν* = 0*,*1*, . . . ,*if*j* = 0. Hence we have*v**j* = 0 if*j* = 0. It follows that*v* =*v*_{0}(*q*_{1}*, p*_{1}).

Moreover, by (3.4) *v* satisﬁes the equation

*−*4*σp*_{1} *∂v*

*∂p*_{1} +*q*_{1} *∂v*

*∂q*_{1} = 0*.*

If we substitute the expansion of *v* into the equation, then, by simple computations,
we easily see that*v* =*φ*(*p*_{1}*q*_{1}^{4}^{σ}) for some analytic function *φ*(*t*) of one variable. This
proves the assertion.

It follows from (2.2) that *v* =*v*_{0} =*φ*(*p*_{1}*q*_{1}^{4}^{σ}) =*φ*(*H|**q*_{2}=0). We deﬁne
*g*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) := *u*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*−φ*(*H*)*.*

(3.6)

By (3.2) and by recalling that*H*is a ﬁrst integral we see that*g* is an analytic solution
of (3.1) such that *g*(*q*_{1}*,*0*, p*_{1}*, p*_{2}) *≡* 0. In order to prove Theorem 2.1 we shall show
*g*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*≡*0 in some neighborhood of the origin. First we will show that

*g*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) =*φ*_{1}(*p*_{1}*q*^{4}_{1}^{σ})*p*_{2}*q*_{2}+*h*_{2}(*q*_{1}*, p*_{1}*, p*_{2})*q*_{2}^{2}+ ˜*h*_{3}(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*q*^{3}_{2}*,*
(3.7)

for some analytic function*φ*_{1}of one variable and analytic functions*h*_{2}and ˜*h*_{3}. Because
*g* is analytic we have the expansion

*g*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) =*g*_{1}(*q*_{1}*, p*_{1}*, p*_{2})*q*_{2}+*h*_{2}(*q*_{1}*, p*_{1}*, p*_{2})*q*^{2}_{2}+ ˜*h*_{3}(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*q*_{2}^{3}*.*
(3.8)

We substitute (3.8) with*u*=*g* into (3.1) and compare the coeﬃcients of*q*_{2}. By (2.1)
we have

*−*4*σq*^{2}_{1}^{σ}*p*_{1}*∂g*_{1}

*∂p*_{1} + 2*σ*

*p*_{2}*∂g*_{1}

*∂p*_{2} *−g*_{1}

+*q*^{2}_{1}^{σ}^{+1}*∂g*_{1}

*∂q*_{1} = 0*.*

(3.9)

By substituting the expansion *g*_{1}(*q*_{1}*, p*_{1}*, p*_{2}) = ^{∞}_{m}_{=0}*g*_{1}_{,m}(*q*_{1}*, p*_{1})*p*^{m}_{2} into (3.9) and by
comparing the coeﬃcients of *p*^{m}_{2} we obtain

*−*4*σq*_{1}^{2}^{σ}*p*_{1}*∂g*_{1}*,m*

*∂p*_{1} + 2*σ*(*m−*1)*g*_{1}*,m*+*q*_{1}^{2}^{σ}^{+1}*∂g*_{1}*,m*

*∂q*_{1} = 0*.*

(3.10)

If*m*= 1, then we obtain a similar equation as for*v**j* in (3.4). Hence we have*g*_{1}*,m*= 0 if
*m*= 1. It follows that*g*_{1} =*g*_{1}_{,}_{1}*p*_{2}, and*g*_{1}_{,}_{1} satisﬁes the equation*−*4*σp*_{1}^{∂g}_{∂p}^{1,1}

1 +*q*_{1}^{∂g}_{∂q}^{1,1}

1 =

0. By the same argument as in the above, we see that *g*_{1} =*g*_{1}_{,}_{1}*p*_{2} = *φ*_{1}(*p*_{1}*q*_{1}^{4}^{σ})*p*_{2} for
some analytic function *φ*_{1} of one variable. This proves the assertion.

Let us now suppose that

*g*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}) = *φ*_{n}_{−1}(*p*_{1}*q*_{1}^{4}^{σ})*p*^{n}_{2}^{−1}*q*^{n}_{2}^{−1}
(3.11)

+ *h*_{n}(*q*_{1}*, p*_{1}*, p*_{2})*q*_{2}^{n}+ ˜*h*_{n}_{+1}(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2})*q*_{2}^{n}^{+1}*,*

for some *n* *≥* 2, some analytic function *φ*_{n}_{−1} of one variable and analytic functions
*h*_{n}(*q*_{1}*, p*_{1}*, p*_{2}) and ˜*h*_{n}_{+1}(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2}). Then we substitute (3.11) into (3.1) with *u*=*g*
and we compare the coeﬃcients of *q*_{2}^{n}. By (2.1) we have

2*p*^{n}_{2}*σ*(2*σ−*1)*q*_{1}^{6}^{σ}^{−2}*φ*^{}_{n}_{−1}*−*4*a*(*q*_{1}^{2}^{σ})(*q*_{1}^{2}^{σ}+ 1)(*n−*1)*p*_{1}*p*^{n}_{2}^{−2}*φ*_{n}_{−1}
(3.12)

*−* 4*σq*_{1}^{4}^{σ}^{−1}*p*_{1}*∂h*_{n}

*∂p*_{1} + 2*σq*_{1}^{2}^{σ}^{−1}

*p*_{2}*∂h*_{n}

*∂p*_{2} *−nh*_{n}

+*q*^{4}_{1}^{σ}*∂h*_{n}

*∂q*_{1} = 0*.*

By substituting the expansion *h**n*(*q*_{1}*, p*_{1}*, p*_{2}) = ^{∞}_{m}_{=0}*h**n,m*(*q*_{1}*, p*_{1})*p*^{m}_{2} into (3.12) and
by comparing the coeﬃcients of *p*^{n}_{2}^{−2} we obtain

*−*4*σq*_{1}^{4}^{σ}^{−1}*p*_{1}*∂h*_{n,n}_{−2}

*∂p*_{1} *−* 4*a*(*q*_{1}^{2}^{σ})(*q*_{1}^{2}^{σ}+ 1)(*n−*1)*p*_{1}*φ**n**−*1

(3.13)

*−* 4*σq*_{1}^{2}^{σ}^{−1}*h*_{n,n}_{−2}+*q*_{1}^{4}^{σ}*∂h*_{n,n}_{−2}

*∂q*_{1} = 0*.*

We will show that

*h**n,n**−*2 = 0*,* *φ**n**−*1 = 0*.*

(3.14)

If we can prove *φ**n**−*1 = 0, then it follows from (3.13) that *v* := *h**n,n**−*2 satisﬁes a
similar equation as (3.4). Hence, by the same argument as for (3.4) we have*h**n,n**−*2 =
0. In order to show *φ*_{n}_{−1} = 0 we insert the expansions

*φ*_{n}_{−1}(*p*_{1}*q*^{4}_{1}^{σ}) =
*∞*

*k*=0

*φ*_{n}_{−1}_{,k}*p*^{k}_{1}*q*_{1}^{4}^{σk}*,* *h*_{n,n}_{−2}(*q*_{1}*, p*_{1}) =
*∞*
*k*=0

*h*_{n,n}_{−2}_{,k}(*q*_{1})*p*^{k}_{1}
(3.15)

into (3.13) and we compare the coeﬃcients of *p*^{k}_{1}. Then we obtain, for *k* *≥*0

*−*4*σq*_{1}^{4}^{σ}^{−1}*kh**n,n**−*2*,k* *−* 4*σq*^{2}_{1}^{σ}^{−1}*h**n,n**−*2*,k*+*q*_{1}^{4}^{σ}*∂h*_{n,n}_{−2}_{,k}

*∂q*_{1}
(3.16)

= 4*a*(*q*_{1}^{2}^{σ})(*q*_{1}^{2}^{σ}+ 1)(*n−*1)*φ**n**−*1*,k**−*1*q*_{1}^{4}^{σ}^{(}^{k}^{−1)}*,*

where we set*φ*_{n}_{−1}_{,}_{−1} = 0. If we set *q*_{1}= 0 and *k*= 1 in (3.16), then, by *a*(0) = 1, we
obtain 0 = 4(*n−*1)*φ*_{n}_{−1}_{,}_{0}, which implies *φ*_{n}_{−1}_{,}_{0} = 0.

Suppose that*φ*_{n}_{−1}_{,k}_{−1} = 0 for some*k* *≥*2. We divide both sides of (3.16) by*q*^{2}_{1}^{σ}^{−1}.
Then the right-hand side of (3.16) is divisible by *q*_{1}^{N}, *N* = 4*σ*(*k* *−*1) + 1 *−*2*σ* *≥*
2*σ*+ 1. Because the operator *−*4*σkq*_{1}^{2}^{σ} +*q*^{2}_{1}^{σ}^{+1}(*d/dq*_{1}) in the left-hand side of the
equation increases the power of *q*_{1}, it follows that *h**n,n**−*2*,k* is divisible by *q*_{1}^{N}. We set
*h*_{n,n}_{−2}_{,k}(*q*_{1}) = *q*_{1}^{N}*W*(*q*_{1}). Then we have *q*_{1}(*d/dq*_{1})*h*_{n,n}_{−2}_{,k} = *q*^{N}_{1} (*N* +*q*_{1}(*d/dq*_{1}))*W*. It
follows from (3.16) that *W* satisﬁes

(*N* *−*4*σk*)*q*_{1}^{2}^{σ}*W* *−*4*σW* + *q*_{1}^{2}^{σ}^{+1}*dW*
*dq*_{1}
(3.17)

= 4*a*(*q*_{1}^{2}^{σ})(*n−*1)*φ*_{n}_{−1}_{,k}_{−1}(*q*_{1}^{2}^{σ}+ 1)*.*

We set *W* = ^{2}_{j}^{σ}_{=0}^{−1}*q*^{j}_{1}*W**j*(*q*^{2}_{1}^{σ}). Because the right-hand side of (3.17) is a function of
*q*_{1}^{2}^{σ}, *W*_{j} (1*≤j <*2*σ*) satisfy

*q*_{1}^{2}^{σ}(*N* *−*4*σk*+*j*)*W**j**−*4*σW**j* +*q*_{1}^{2}^{σ}^{+1}*dW*_{j}
*dq*_{1} = 0*.*

(3.18)

By a similar argument as for (3.4) we have *W**j* = 0 for 1 *≤* *j <* 2*σ*. Hence we have
*W*(*q*_{1}) = *W*_{0}(*q*^{2}_{1}^{σ}) =: *V*(*t*) (*t* = *q*_{1}^{2}^{σ}). Because *q*_{1}(*d/dq*_{1})*V* = 2*σt*(*d/dt*)*V*, it follows
from (3.17) that

(1*−*6*σ*)*tV* *−* 4*σV* + 2*σt*^{2}*dV*

*dt* = 4*a*(*t*)(*n−*1)(*t*+ 1)*φ**n**−*1*,k**−*1*.*

If we set *U* := *V /*(2(*n* *−*1)*φ*_{n}_{−1}_{,k}_{−1}), then *U* is an analytic solution of (2.4). This
contradicts to the assumption of the theorem, because we assume that (M) is not
veriﬁed. Hence we have *φ**n**−*1*,k**−*1 = 0. Because *k* is arbitrary we have *φ**n**−*1 = 0.

Next we set *φ**n**−*1 = 0 in (3.12) and consider the coeﬃcients of *p*^{m}_{2} (*m* =*n*). Then
we see that *h**n,m* satisﬁes a similar equation as for (3.4). Hence we have *h**n,m* = 0
if *n* = *m*, and *h**n,n* = *φ**n*(*p*_{1}*q*_{1}^{4}^{σ}) for some analytic function *φ**n* of one variable. It
follows that *h**n*(*q*_{1}*, p*_{1}*, p*_{2}) = *h**n,n*(*q*_{1}*, p*_{1})*p*^{n}_{2} =*φ**n*(*p*_{1}*q*_{1}^{4}^{σ})*p*^{n}_{2}. Hence we have (3.11) with
*n* replaced by *n*+ 1. By induction we obtain (3.11) for an arbitrary integer*n* *≥*2.

It follows from (3.11) with *n* replaced by *n* + 2 that, for every *n* *≥* 0 we have

*∂*_{q}^{n}_{2}*g*(*q*_{1}*,*0*, p*_{1}*, p*_{2}) *≡* 0, where (*q*_{1}*, p*_{1}*, p*_{2}) is in some neighborhood of the origin which
may depend on*n*. On the other hand *∂*_{q}^{n}_{2}*g*(*q*_{1}*,*0*, p*_{1}*, p*_{2}) is analytic in some neighbor-
hood of the origin independent of*n*. By analytic continuation, we have*∂*_{q}^{n}_{2}*g*(*q*_{1}*,*0*, p*_{1}*, p*_{2})

*≡*0 in some neighborhood of the origin independent of *n*. By the partial Taylor ex-
pansion*g* = _{n}*∂*_{q}^{n}_{2}*g*(*q*_{1}*,*0*, p*_{1}*, p*_{2})*q*_{2}^{n}*/n*!, we have *g* = 0.

4. Transseries expansion of first integral

In this section, we shall construct a ﬁrst integral of (2.3) as a transseries. In order
to introduce such a series we consider the terms in (3.1) which preserve the order of
*q*_{2}

*Lu*:=*q*_{1}^{2}^{σ}^{−1}

2*σ*(*p*_{2} *∂u*

*∂p*_{2} *−q*_{2}*∂u*

*∂q*_{2}) +*q*^{2}_{1}^{σ}(*q*_{1}*∂u*

*∂q*_{1} *−*4*σp*_{1}*∂u*

*∂p*_{1})

*.*
(4.1)

We note that we can write (3.1) in the form

*Lu*+*Ru* = 0*,* *Ru* :=*{H, u} − Lu.*

(4.2)

In order to construct an inverse of *L* we consider the Lagrange-Charpit system cor-
responding to *L*

*dq*_{1}

*q*^{4}_{1}^{σ} = *dq*_{2}

*−*2*σq*_{1}^{2}^{σ}^{−1}*q*_{2} = *dp*_{1}

*−*4*σq*^{4}_{1}^{σ}^{−1}*p*_{1} = *dp*_{2}
2*σq*^{2}_{1}^{σ}^{−1}*p*_{2}*.*
(4.3)

We integrate (4.3) by taking *q*_{1} as an independent variable. By simple computations
we can easily see that the solution of (4.3) is given by

*q*_{2} =*q*_{2}^{0}exp
*q*_{1}^{−2}^{σ}

*, p*_{2} =*p*^{0}_{2}exp

*−q*^{−2}_{1} ^{σ}

*, p*_{1} =*p*^{0}_{1}*q*_{1}^{−4}^{σ}*,*
(4.4)

where *q*^{0}_{2}, *p*^{0}_{2} and *p*^{0}_{1} are certain constants.

We note that the solution of the homogeneous equation *Lv* = 0 is given by
*v* =*φ*(*p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp

*q*_{1}^{−2}^{σ}

*, q*_{2}exp

*−q*_{1}^{−2}^{σ}
)*,*
(4.5)

with *φ*(*p*^{0}_{1}*, p*^{0}_{2}*, q*_{2}^{0}) being an arbitrary function of *p*^{0}_{1}, *p*^{0}_{2} and *q*^{0}_{2}. We then construct a
solution*u* of (4.2) in the form

*u*=
*∞*

*j*=0

*u*_{j}(*q*_{1}*, p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp
*q*_{1}^{−2}^{σ}

) exp

*−q*_{1}^{−2}^{σ}
*q*_{2}*j*

*,*
(4.6)

where*u*_{0}(*q*_{1}*, p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp
*q*_{1}^{−2}^{σ}

)*≡u*_{0}(*p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp
*q*_{1}^{−2}^{σ}

). We call (4.6) the transseries solution of (4.2). Then we have

Proposition 4.1**.** *Let* *u*_{0}(*p*^{0}_{1}*, p*^{0}_{2}) *be a given analytic function of* *p*^{0}_{1} *and* *p*^{0}_{2} *such that*

*∂u*_{0}*/∂p*^{0}_{2} = 0*. Then (2.3) is formally Liouville-integrable in the sense that (4.6) is a*
*formal integral of (2.3) which is functionally independent of* *H.*

*Proof.* We note that *R* in (4.2) has analytic coeﬃcients and *R* raises the power of
*q*_{2} at least by one. On the other hand we have

*L*
*u*_{j}

exp

*−q*_{1}^{−2}^{σ}
*q*_{2}*j*

= (*Lu*_{j})
exp

*−q*_{1}^{−2}^{σ}
*q*_{2}*j*

*.*
(4.7)

Hence, if we substitute (4.6) into (4.2) and compare the coeﬃcients of *q*^{j}_{2} of both
sides, then we have

*Lu**j* = ( linear functions of *u**k* and their derivatives (*k < j*)*,* *j* = 1*,*2*, . . .*
(4.8)

We note that the right-hand side is a known quantity if we determine *u**j* recursively.

We will solve*Lv* =*f*, where
*v* =*v*

*q*_{1}*, p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp
*q*_{1}^{−2}^{σ}

*.*

By making the change of variables (*q*_{1}*, p*_{1}*, p*_{2})*→*(*q*_{1}*, p*^{0}_{1}*, p*^{0}_{2}) given by (4.4), the equa-
tion *Lv* =*f*(*q*_{1}*, p*_{1}*, p*_{2}) is written in the form

*q*^{4}_{1}^{σ}(*∂v/∂q*_{1}) =*g*(*q*_{1}*, p*^{0}_{1}*, p*^{0}_{2})*,*
(4.9)

where

*g* *≡g*(*q*_{1}*, p*^{0}_{1}*, p*^{0}_{2}) =*f*(*q*_{1}*, p*^{0}_{1}*q*_{1}^{−4}^{σ}*, p*^{0}_{2}exp

*−q*_{1}^{−2}^{σ}
)*.*

Hence the solution of (4.9) is given by
*v* =

_{q}_{1}

*a*_{0}

*s*^{−4}^{σ}*g*(*s, p*^{0}_{1}*, p*^{0}_{2})*ds,*
(4.10)

where*a*_{0} is an arbitrary complex constant. If we go back to the original variables *q*_{1},
*p*_{1} and *p*_{2}, then we obtain a solution of *Lv* = *f*. Therefore we have a solution *u* of
(4.2) given by (4.6).

Finally, we will show that *u* converges, then *u* is an integral of (2.3) functionally
independent of *H*. Hence our Hamiltonian system is formally Liouville-integrable.

Indeed, if this is not the case, then we have*u*=*φ*(*H*) for some smooth function *φ* of
one variable. If we set *q*_{2}= 0, then we obtain

*u*_{0}

*p*_{1}*q*_{1}^{4}^{σ}*, p*_{2}exp
*q*_{1}^{−2}^{σ}

= *φ*(*H*)*|*_{q}_{2}_{=0} =*φ*(*p*_{1}*q*_{1}^{4}^{σ})*.*

This is a contradiction to the assumption that *∂u*_{0}*/∂p*^{0}_{2} = 0. This ends the proof.

5. Convergence of transseries

In this section we consider the Hamiltonian corresponding to*r* =*q*_{1}^{2}^{σ}+*a*(*q*_{1}^{2}^{σ})*q*_{2}^{2} in
(2.2)

*H* =*−*2*σq*^{2}_{1}^{σ}^{−1}*q*_{2}*p*_{2}(1 +*q*_{2}^{2}*a*^{}) +*p*_{1}

(*q*_{1}^{2}^{σ}+*aq*_{2}^{2})^{2}+ 2*aq*^{2}_{2}
*,*
(5.1)

where we assume *a*(0) = 1 for the sake of simplicity. We study the convergence of
transseries solutions (4.6), where we set *a*_{0} = 0 in (4.10). Note that *u*_{j}(0*, p*^{0}_{1}*, p*^{0}_{2})
identically vanishes for any *j* *≥*1.

Clearly, the integral*u*is the solution of (4.2), where*L*and*R*are given, respectively,
by (4.1) and

*Ru* = ( ˜*αp*_{1}+ ˜*βp*_{2})*∂u*

*∂p*_{1} +

˜

*γp*_{2}+ ˜*δp*_{1}
*∂u*

*∂p*_{2}
(5.2)

+ *E*(*q*_{1}*, q*_{2})*∂u*

*∂q*_{1} *−*2*σq*^{2}_{1}^{σ}^{−1}*a*^{}*q*_{2}^{3}*∂u*

*∂q*_{2}*,*
where

*E* *≡* *E*(*q*_{1}*, q*_{2}) := *aq*_{2}^{2}(*aq*^{2}_{2}+ 2*q*_{1}^{2}^{σ} + 2)*,*
(5.3)

˜

*α* := *−*4*σq*^{2}_{1}^{σ}^{−1}*q*^{2}_{2}(*a*+*a*^{}+*a*^{}*q*_{1}^{2}^{σ} +*aa*^{}*q*_{2}^{2})*,*
(5.4)

*β*˜ := 2*σ*(2*σ−*1)*q*_{2}*q*_{1}^{2}^{σ}^{−2}(1 +*a*^{}*q*_{2}^{2}) + 4*σ*^{2}*q*^{4}_{1}^{σ}^{−2}*a*^{}*q*_{2}^{3}*,*
(5.5)

˜

*γ* := 6*σq*_{1}^{2}^{σ}^{−1}*a*^{}*q*_{2}^{2}*,* *δ*˜:=*−*4*q*_{2}*a*(*q*_{1}^{2}^{σ}+*aq*_{2}^{2}+ 1)*.*

(5.6)

Let*ε*_{0} be a small positive constant. Then we deﬁne
*S*_{0}:=*{q*_{1} *∈*C;*|q*_{1}*|< ε*_{0}*} ∩*

*q*_{1} *∈*C; Re*q*_{1}^{2}^{σ} *<*0
*,*
(5.7)

where Re*q*_{1}^{2}^{σ} denotes the real part of *q*_{1}^{2}^{σ}. Then we have

Theorem 5.1**.** *Letu*_{0}(*p*^{0}_{1}*, p*^{0}_{2})*be an analytic function ofp*^{0}_{1} *andp*^{0}_{2} *such that∂u*_{0}*/∂p*^{0}_{2} =
0 *in some neighborhood of the origin. Then there exist a* *δ >*0*, an* *ε*_{0} *>*0*, neighbor-*
*hoods* *V*_{1}*,* *V*_{2} *of the origin in* C *such that* *u* *in (4.6) converges in the domain*

*{*(*q*_{1}*, q*_{2}*, p*_{1}*, p*_{2});*q*_{1}*∈S*_{0}*, p*_{1} *∈V*_{1}*, p*_{2} *∈V*_{2}*,|*exp

*−q*_{1}^{−2}^{σ}

*q*_{2}*|< δ}.*

*Proof.* Let*L*be given by (4.1). The Lagrange-Charpit system corresponding to*L*+*R*
is given by

*dq*_{1}

*q*^{4}_{1}^{σ}+*E* = *dq*_{2}

*−*2*σq*_{1}^{2}^{σ}^{−1}*q*_{2}*T* = *dp*_{1}

( ˜*α−*4*σq*_{1}^{4}^{σ}^{−1})*p*_{1}+ ˜*βp*_{2} = *dp*_{2}

(2*σq*^{2}_{1}^{σ}^{−1}+ ˜*γ*)*p*_{2} + ˜*δp*_{1}*,*
(5.8)

where we set

*T* *≡T*(*q*_{1}*, q*_{2}) := 1 +*a*^{}*q*_{2}^{2}*.*
(5.9)

We integrate (5.8) by taking *q*_{1} *∈* *S*_{0} as an independent variable. We want to show
that the solutions are perturbations of the solutions (4.4) in *S*_{0}. Namely, we will
prove

*q*_{2} = *q*^{0}_{2}exp(*q*^{−2}_{1} ^{σ}) exp(*q*_{1}*Q*_{2}(*q*_{1}*, q*_{2}^{0}))*,*
(5.10)

*p*_{1} = *q*^{−4}_{1} ^{σ}

*p*^{0}_{1}(1 +*P*_{1}(*q*_{1}*, q*_{2}^{0})) +*p*^{0}_{2}*P*˜_{1}(*q*_{1}*, q*_{2}^{0})

*,*
(5.11)

*p*_{2} = exp(*−q*_{1}^{−2}^{σ})

*p*^{0}_{1}*P*_{2}(*q*_{1}*, q*^{0}_{2}) +*p*^{0}_{2}(1 + ˜*P*_{2}(*q*_{1}*, q*_{2}^{0}))

*,*
(5.12)

for some functions *Q*_{2} *≡* *Q*_{2}(*q*_{1}*, q*_{2}^{0}), *P**j* *≡* *P**j*(*q*_{1}*, q*_{2}^{0}) and ˜*P**j* *≡* *P*˜*j*(*q*_{1}*, q*_{2}^{0}) (*j* = 1*,*2),
which are holomorphic and bounded when *q*_{1} *∈* *S*_{0} and *q*_{2}^{0} in some neighborhood of
*q*_{2}^{0} = 0. Here*p*^{0}_{1} and *p*^{0}_{2} are arbitrary constants.

In order to verify these properties we ﬁrst consider the following equation
*dq*_{2}

*dq*_{1} = *−*2*σq*_{2}*q*_{1}^{2}^{σ}^{−1}(1 +*a*^{}*q*_{2}^{2})

*q*^{4}_{1}^{σ}+*E* = *−*2*σq*_{2}
*q*_{1}^{2}^{σ}^{+1}

1 + *a*^{}*q*^{2}_{2}*−q*_{1}^{−4}^{σ}*E*
1 +*q*^{−4}_{1} ^{σ}*E*

*.*
(5.13)

Clearly, *q*_{2} = 0 is a solution of (5.13). We assume *q*_{2} *≡* 0. We note that *v* :=

*q*_{2}^{0}exp(*q*_{1}^{−2}^{σ}) satisﬁes the equation *dv/dq*_{1} =*−*2*σvq*_{1}^{−2}^{σ}^{−1}. We set *U* :=*q*_{1}*Q*_{2}, and we
substitute (5.10) into (5.13). Then we have

*dU*

*dq*_{1} =*−*2*σ*(*a*^{}*q*_{2}^{2}*−q*^{−4}_{1} ^{σ}*E*)

*q*_{1}^{2}^{σ}^{+1}(1 +*q*_{1}^{−4}^{σ}*E*) =:*f*(*q*_{1}*, U*)*,*
(5.14)

where *q*_{2} = *q*_{2}^{0}exp(*q*_{1}^{−2}^{σ})*e*^{U}. Because Re*q*^{2}_{1}^{σ} *<* 0 in *S*_{0}, we see that *f*(*q*_{1}*, U*) is holo-
morphic when (*q*_{1}*, U*) *∈* *S*_{0} *×* Ω, and continuous up to its closure, where Ω is a
neighborhood of the origin. Moreover, its maximal norm when (*q*_{1}*, U*)*∈* *S*_{0} *×*Ω can
be made arbitrarily small if we shrink *S*_{0} suﬃciently small.

We will solve (5.14) in*S*_{0}. If we replace *U* with *U* +*c* for a constant*c* we see from
(5.10) that *q*_{2}^{0} is replaced by *e*^{c}*q*_{2}^{0}. Hence we may assume that *U* vanishes at *q*_{1} = 0.

We will look for the solution *U* as the solution of the following equation
*U* =

_{q}_{1}

0

*f*(*s, U*)*ds,*
(5.15)

where the integral is taken along the straight line in*S*_{0} which connects 0 and*q*_{1}. We
note that we can make*|f*(*q*_{1}*, U*)*|*arbitrarily small if we take*ε*_{0} in*S*_{0} suﬃciently small
and *U* is bounded. We can easily show that the right-hand side operator of (5.15) is
a contraction mapping on a small ball in the set of functions holomorphic in *S*_{0} and
continuous on its closure, and *U*(0) = 0. Hence we have a holomorphic solution *U* in
*S*_{0} of (5.15). If we set*Q*_{2} :=*q*^{−1}_{1} *U* we obtain the desired solution. The analyticity of
*U* with respect to *q*_{2}^{0} is easy to verify because of the deﬁnition of *f*.

Next we study the equations for *p*_{1} and *p*_{2}. It follows from (5.8) that
*dp*_{1}

*dq*_{1} =*αp*_{1} +*βp*_{2}*,* *dp*_{2}

*dq*_{1} =*γp*_{2} +*δp*_{1}*,*
(5.16)

where

*α*= *α*˜*−*4*σq*_{1}^{4}^{σ}^{−1}
*q*_{1}^{4}^{σ}+*E* *, β*=

*β*˜

*q*_{1}^{4}^{σ}+*E, γ* = 2*σq*_{1}^{2}^{σ}^{−1}+ ˜*γ*
*q*^{4}_{1}^{σ}+*E* *, δ* =

*δ*˜
*q*^{4}_{1}^{σ}+*E.*
(5.17)

We will construct the solution of (5.16) in the following form
*p*_{ν} =

*∞*
*j*=0

*p*^{(}_{ν}^{j}^{)}*,* *ν*= 1*,*2*,*
(5.18)

where

*dp*^{(0)}_{1}

*dq*_{1} =*αp*^{(0)}_{1} *,* *dp*^{(0)}_{2}

*dq*_{1} =*γp*^{(0)}_{2} *,*
(5.19)

and *p*^{(}*ν*^{j}^{)} (*ν* = 1*,*2;*j* = 1*,*2*, . . .*) are determined by
*dp*^{(}_{1}^{j}^{)}

*dq*_{1} =*αp*^{(}_{1}^{j}^{)}+*βp*^{(}_{2}^{j}^{−1)}*,* *dp*^{(}_{2}^{j}^{)}

*dq*_{1} =*γp*^{(}_{2}^{j}^{)}+*δp*^{(}_{1}^{j}^{−1)}*.*
(5.20)

First we solve (5.19). By the deﬁnition of *γ* in (5.17) we have the expression
*γ* = 2*σq*_{1}^{−2}^{σ}^{−1}+*γ*_{0}, where*γ*_{0} is a bounded holomorphic function in*S*_{0}. By the change
of an unknown function similar in the argument for (5.13) the solution *p*^{(0)}_{2} of (5.19)
has the following expression

*p*^{(0)}_{2} =*p*^{0}_{2}exp(*−q*^{−2}_{1} ^{σ})(1 + ˜*P*_{2}^{(0)}(*q*_{1}*, q*_{2}^{0}))
(5.21)

for some ˜*P*_{2}^{(0)} which is bounded and holomorphic in *q*_{1} *∈* *S*_{0} and *q*_{2}^{0} in some neigh-
borhood of the origin, where *p*^{0}_{2} is an arbitrary constant. Similarly, noting that
*α* = *−*4*σq*_{1}^{−1}+*α*_{0} for some bounded holomorphic function *α*_{0} in *S*_{0} we see that the
solution*p*^{(0)}_{1} of (5.19) has the following expression

*p*^{(0)}_{1} =*p*^{0}_{1}*q*^{−4}_{1} ^{σ}(1 + ˜*P*_{1}^{(0)}(*q*_{1}*, q*_{2}^{0}))
(5.22)

for some ˜*P*_{1}^{(0)} which is bounded and holomorphic in *q*_{1} *∈S*_{0} and*q*_{2}^{0} in some neighbor-
hood of the origin, where*p*^{0}_{1} is an arbitrary constant.

Now we assume that*p*^{(}*ν*^{k}^{)}’s (*k* = 0*,*1*, . . . , j−*1,*ν* = 1*,*2)