ANALYTIC INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH EXCEPTIONAL POTENTIALS

We study the existence of analytic first integrals of the complex Hamiltonian systems of the form


Introduction and statement of the main results
Ordinary differential equations in general and Hamiltonian systems in particular play a very important role in many branches of the applied sciences. The question whether a differential system admits a first integral is of fundamental importance as first integrals give conservation laws for the model and that enables to lower the dimension of the system. Moreover knowing a sufficient number of first integrals allows to solve the system explicitly. Until the end of the 19th century the majority of scientists thought that the equations of classical mechanics were integrable and finding the first integrals was mainly a problem of computation. In fact, now we know that the integrability is a nongeneric phenomenon inside the class of Hamiltonian systems (see [3]), and in general it is very hard to determine whether a given Hamiltonian system is integrable or not.
In this work we are concerned with the integrability of the natural Hamiltonian systems defined by a Hamiltonian function of the form where V (q 1 , q 2 ) ∈ C[q 1 , q 2 ] is a homogeneous polynomial potential of degree k. As usual C[q 1 , q 2 ] is the ring of polynomial functions over C in the variables q 1 and q 2 .
To be more precise we consider the following system of four differential equations Let A = A(q, p) and B = B(q, p) be two functions, where p = (p 1 , p 2 ) and q = (q 1 , q 2 ). We define the Poisson bracket of A and B as  (2) is completely or Liouville integrable if it has 2 functionally independent first integrals H and F . As usual H and F are functionally independent if their gradients are linearly independent at all points of C 4 except perhaps in a zero Lebesgue measure set. Let PO 2 (C) denote the group of 2 × 2 complex matrices A such that AA T = α Id, where Id is the 2 × 2 identity matrix and α ∈ C \ {0}. The potentials V 1 (q) and V 2 (q) are equivalent if there exists a matrix A ∈ PO 2 (C) such that V 1 (q) = V 2 (Aq). Therefore we divide all potentials into equivalent classes. In what follows a potential means a class of equivalent potentials in the above sense. This definition of equivalent potentials is motivated by the following simple observation (for a proof see [1]). Let V 1 and V 2 be two equivalent potentials. If the Hamiltonian system (2) with the potential V 1 is integrable, then it is also integrable with the potential V 2 .
It was shown in [2] that among all equivalent potentials one can always choose a representative V such that the polynomial V has one root in an arbitrary point of CP 1 \ {[1 : +i], [1 : −i]}. This is always possible except for cases when all linear factors of V have the form q 2 ± iq 1 , that is, if the potential V is of the form These potentials are called exceptional. It was proved in [1] that the exceptional potentials V 0 , V 1 , V k−1 , V k and V k/2 when k is even are integrable . It is easy to find that for these exceptional potentials the additional polynomial first integral is: and when k is even I k/2 = q 2 p 1 − q 1 p 2 . It is also claimed in [2] and [1] that nothing is known about the integrability of the remaining exceptional potentials. In this paper we focus on these remaining exceptional potentials. We restrict to the potentials V l with l = 2, . . . , k/2 − 1, k/2 + 1, . . . , k − 2 and k even. Note that if k ≤ 4 all the exceptional potentials are integrable with polynomial first integrals. So we will focus on the case k ≥ 5.
System (2) becomeṡ Our main results are the following. The proof of Theorem 1 is given in section 3. We state the following conjecture. In the case in which l = 2 or l = k − 2 with k being either even or odd, we can also prove with different techniques the non-existence of rational first integrals. The proof of Theorem 2 is given in section 2.
2. Weight-homogeneous polynomial differential system and proof of Theorem 2 We consider polynomial differential system of the form x 4 ] for i = 1, 2, 3, 4. As usual N, R and C denote the sets of positive integers, real and complex numbers, respectively; and C[x 1 , x 2 , x 3 , x 4 ] denotes the polynomial ring over C in the variables x 1 , x 2 , x 3 , x 4 . Here t can be real or complex. We say that system (4) is weight-homogeneous if there exist s = (s 1 , s 2 , s 3 , s 4 ) ∈ N 4 and d ∈ N such that for arbitrary a ∈ R + = {a ∈ R, a > 0} we have for i = 1, 2, 3, 4. We call s = (s 1 , s 2 , s 3 , s 4 ) as the weight exponent of system (4) and d as weight degree with respect to the weight exponent s.
We say that a polynomial F ( The following well-known proposition (easy to prove) reduces the study of the existence of analytic first integrals of a weight-homogeneous polynomial differential system (4) to the study of the existence of a weight-homogeneous polynomial first integrals. Proposition 3. Let H be an analytic function and let H = ∑ i H i be its decomposition into weight-homogeneous polynomials of weight degree i with respect to the weight exponent s. Then H is an analytic first integral of the weight-homogeneous polynomial differential system (4) with weight exponent s if and only if each weight-homogeneous part H i is a first integral of system (4) for all i.
We introduce the change of variables In these new variables system (3) becomeṡ (5)
From Proposition 3 and the observation above it follows that for proving the existence of non-existence of analytic first integrals of system (5) it is sufficient to show the existence or non-existence of weighthomogeneous polynomial first integrals with weight exponents given in (6).
We recall that in the case in which k is even, we can be more precise and it is clear that system (5) is a weight-homogeneous polynomial differential system with weight exponent (1, 1, k/2, k/2) and weight degree d = k/2.
Proof of Theorem 2. Instead of Proving Theorem 2 we will prove the following theorem which is equivalent to Theorem 2.
Theorem 4. System (5) with l = 2 or with l = k − 2 does not admit an additional rational first integral.
We will only prove the case l = k − 2 because the proof of the case l = 2 is exactly the same interchanging the roles of y 1 with y 2 and of x 1 with x 2 . The proof follows directly from the following theorem which is Theorem 2.4 in [2].

Proof of Theorem 1
In this section we will prove the following equivalent result to Theorem 1.
We first observe that we only need to prove Theorem 6 for the cases l = 2, . . . , k 2 − 1, because the proof of the cases l = k 2 + 1, . . . , k − 2 is exactly the same interchanging the roles of x 1 with x 2 , and y 1 with y 2 .
Before going into the technicalities of the proof of Theorem 6, we would like to highlight the main idea behind the proof. First we shall restrict system (5) to the zero level of the first integral H, which is a polynomial function. The restriction to this level set gives rise to a nontrivial rational first integralF of the restricted system. To be more precise,F (y 1 , y 2 , x 1 ) is a polynomial in the variables y 1 , y 2 , x 1 and x −1 1 . So, it can be written in the following form: We recall again that system (5) is a weight-homogeneous polynomial differential system with weight exponent (1, 1, k/2, k/2) and weight degree d = k/2. From section 3 it follows that for proving Theorem 6 it is sufficient to show that this system has no weight-homogeneous polynomial first integrals with weight exponent (1, 1, k/2, k/2).
Let F = F (y 1 , y 2 , x 1 , x 2 ) ∈ C[y 1 , y 2 , x 1 , x 2 ] be a weight-homogeneous polynomial first integral of system (5) with weight exponent (1, 1, k/2, k/2) and weight degree d = k 2 n with n ≥ 1. We can express it as The function F cannot depend only on y 1 and y 2 . Indeed, if F = F (y 1 , y 2 ) then from (5) we get and consequently F is a constant. So F depends on x 1 or x 2 , and thus n ≥ 2.
We study the first integral F on the level set H = 0 by eliminating, for example x 2 as follows: Thus, we end up with the following system: Note that the restriction of the polynomial first integral F to the level set H = 0 can be written as where eachF j (y 1 , y 2 ) is a homogeneous polynomial of weight degree M := k 2 (n − j). Indeed, the degree ofF j (y 1 , y 2 ) is l 1 + l 2 + ll 4 + (k − l)l 4 = l 1 + l 2 + kl 4 .
using that l 1 + l 2 = n − k 2 (l 3 + l 4 ) and l 3 − l 4 = j we can rewrite the above expression as

Note that system (8) is completely integrable with the first integrals
Using thatF must satisfy (9), we must have that for any m = 0, . . . , j 2 , which yields ) .
Note thatF must be a polynomial in the variables y 1 and y 2 . Thus This implies that Moreover using again (9) we have that j 1 ≤ n. Therefore, with β j 1 ,n,k,l ∈ C.
To conclude the proof of Theorem 1 it is sufficient to show that F = 0. Indeed ifF = 0 then any weight homogenous polynomial first integral with weight exponent (1, 1, k/2, k/2) and weight degree d = kn/2 restricted to H = 0 is zero and thus system (5) cannot have a weight homogenous polynomial first integral F with weight exponent (1, 1, k/2, k/2) and weight degree d = kn/2 independent with H since otherwise when restricted to H = 0 this first integral would not be zero.