ON THE POLYNOMIAL INTEGRABILITY OF A SYSTEM MOTIVATED

. We consider diﬀerential systems obtained by coupling two Euler–Poinsot systems. The motivation to consider such systems can be traced back to the Riemann ellipsoid problem. We provide new cases for which these systems are completely integrable. We also prove that these systems either are completely integrable or have at most four functionally independent analytic ﬁrst integrals.


Introduction and statement of the main results
Consider the following system of differential equations dx dt = ∇ x G(x, y) ∧ x, dy dt = ∇ y G(x, y) ∧ y, (1.1) where (x, y) ∈ R 3 × R 3 and G is the quadratic form (a i (x 2 i + y 2 i ) + 2b i x i y i ).
Expanding the notation, system (1.1) writes as: 3 , y 1 , y 2 , y 3 ), 3 , y 1 , y 2 , y 3 ), 3 , y 1 , y 2 , y 3 ), 3 x 3 y 2 + (a 2 − a 3 )y 2 y 3 = P 4 (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ), y 2 = b 3 x 3 y 1 − b 1 x 1 y 3 + (a 3 − a 1 )y 1 y 3 = P 5 (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ), a 2 )y 1 y 2 = P 6 (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ), (1.2) where (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) ∈ R 6 and a i , b i ∈ R for i = 1, 2, 3 such that at least one b i is assumed to be different from zero. It is immediate to verify that system (1.2) has the following polynomial three first integrals which are functionally independent. We recall that given U an open set of R 6 such that R 6 \U has zero Lebesgue measure, we say that a real function H = H(x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ): U ⊂ R 6 → R is a first integral if H is constant for all values of a solution (x 1 (t), x 2 (t), x 3 (t), y 1 (t), y 2 (t), y 3 (t)) of system (1.2) contained in U , i.e. H is a first integral in U if and only if 3 , y 1 , y 2 , y 3 ) = 0 on the points of U . Moreover, the first integrals H 1 , . . . , H r are functionally independent if the r × 6 matrix ⎛ ⎜ ⎝ has rank r at all points (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) ∈ R 6 where they are defined except perhaps in a zero Lebesgue measure set. We are interested in finding additional polynomial first integrals which are functionally independent with H 1 , H 2 and H 3 . We recall that the first natural class to investigate the existence of first integrals is the class of the polynomials since it is the easiest one. Only after this class is completely understood it makes sense to go to bigger classes of functions to look for integrability.
We know that since system (1.2) has zero divergence it follows from Theorem 2.7 of [5] that if it has 4 functionally independent analytic first integrals then the system is completely integrable, i.e. it has 5 first integrals functionally independent. We note that system (1.2) is invariant under the diffeomorphism First we obtain some polynomial first integrals. (b) b 1 = ±b 3 and a 1 = a 3 , then H 4 = ±x 2 + y 2 ; (c) b 2 = ±b 3 and a 2 = a 3 , then H 4 = ±x 1 + y 1 ; The cases of integrability of Theorem 1.1 were already known by Negrini (see Thms. 2 and 3 of [9]), but he did not know that the fourth functionally independent first integral of systems (1. Corollary 1.2 is proved in Section 2, but it was also known by Negrini in [9]. In [9] the author also gives conditions for the existence or non-existence of meromorphic first integrals for system (1.2). Theorem 1.3. If a differential system (1.2) is completely integrable with analytic first integrals, then it satisfies one of the conditions of Theorem 1.1. Otherwise, the differential system (1.2) has at most 4 functionally independent analytic first integrals. Theorem 1.3 is proved in Section 3.

Proof of Corollary 1.2
The following result is due to Jacobi. For a proof in a more general setting see Theorem 2.7 of [5]. Theorem 2.1. Consider an analytic differential system in R n of the form with P (x) = (P 1 (x), . . . , P n (x)). Assume that

it has zero divergence)
and that it admits n − 2 first integrals, I i (x) = c i with i = 1, . . . , n − 2 functionally independent. These integrals define, up to a relabeling of the variables, an invertible transformation mapping from (x 1 , . . . , x n ) to (c 1 , . . . , c n−2 , x n−1 , x n ) given by Let Δ be the Jacobian of the transformation Then system (2.1) admits an extra first integral given by where the tilde denotes the quantities expressed in the variables (c 1 , . . . , c n−2 , x n−1 , x n ). Moreover this first integral is functionally independent with the previous n − 2 first integrals, that is, the system is completely integrable.
Note that from the expression of the additional first integral I n−1 , we see that in general such extra first integral can be non-analytic when the previous n − 2 first integrals are analytic.
Proof of Corollary 1.2. It is immediate to verify that the differential systems (1.2) in R 6 have zero divergence because every P i does not depend on x i for i = 1, 2, 3, and P i does not depend on y i−3 for i = 4, 5, 6. In the case of the conditions given in Theorem 1.1 the differential systems (1.2) have 4 = 6 − 2 first integrals functionally independent. So in this case they satisfy the assumptions of Theorem 2.1. Therefore this case is completely integrable.

Proof of Theorem 1.3
We denote by Z + the set of non-negative integers. The following result, due to Zhang [12], will be used in a strong way in the proof of Theorem 1.3.

Theorem 3.1.
For an analytic vector field X defined in a neighborhood of the origin in R n associated to system (2.1) with P (0) = 0, let λ 1 , . . . , λ n be the eigenvalues of DP (0). Set Assume that system (2.1) has r < n functionally independent analytic first integrals Φ 1 (x), . . . , Φ r (x) in a neighborhood of the origin. If the Z-linear space generated by G has dimension r, then any nontrivial analytic first integral of system (2.1) in a neighborhood of the origin is an analytic function of Φ 1 (x), . . . , Φ r (x).
Extensions of Theorem 3.1 can be found in [4,7]. We call each element (k 1 , . . . , k n ) ∈ G a resonant lattice of the eigenvalues λ 1 , . . . , λ n . Direct calculations show that the differential systems (1.2) have seven planes of singularities, but we only use for proving our result two of these planes of singularities.
The idea of the proof is to use Theorem 3.1. Note that From Theorem 3.1 we know that the number of functionally independent analytic first integrals of the differential systems (1.2) in a neighborhood of each of the singularities, say for example, S 1 is at most the number of linearly independent elements of the set So, we compute when we can have five linearly independent elements of the set G 1 . We find that to have five linearly independent elements we must satisfy two type of conditions that we call (i) and (ii). Investigating condition (i) we find some conditions given in Theorem 1.1 and some others. To study conditions (i) that are not inside the condition given in Theorem 1.1 we use a second lattice G 2 corresponding to a second singularity S 2 of the system. We compute again, for S 2 when, under these remaining conditions the number of linearly independent elements of the set G 2 can be five and we find that this is only possible when we are under some of the conditions given in Theorem 1. We compute again, for S 2 when, under these extra conditions the number of linearly independent elements of the set G 2 can be five and we find that never. So the proof will be complete.
In order that equation (3.3) has an additional linearly independent non-negative integer solutions different from the above list, we must have: This last equality can be written in an equivalent way as where we have used the fact that B 1 = A 2 1 − 4Δ 1 . In case (i) we obtain the following independent conditions: In the first four cases we are inside the conditions of Theorem 1.1. Now we shall consider the last four cases. We denote them by Proof. At the singularity S 2 = (x 1 , 0, 0, y 1 , 0, 0), the 6-tuple of eigenvalues of the linear part of the differential systems (1.2) are given by This last equation can be written as Clearly we have that 3 )x 1 y 1 . Without loss of generality we can write it as and it is easy to check that Then, from (3.7) and (3.8) equating to zero the coefficients of the monomials in the variables x 1 and y 1 we have where L 2 /K 2 = m 2 n 2 /(n 2 2 + m 2 2 ) = 0. For the conditions s 1 and s 2 we have that the solutions of (3.9) are where the last condition corresponds in fact to four conditions. In all cases we are under the assumptions of Theorem 1.1 (note that in the last four cases we have in fact that b 3 = ±(a 2 − a 3 )). Finally, for the conditions s 3 and s 4 we have that the solutions of (3.9) are where every one of the two last conditions correspond in fact to two conditions. In all cases we are under the assumptions of Theorem 1.1 (note that in the last four cases we also have that b 3 = ±(a 2 − a 3 )). This ends the proof of the lemma.
From Theorem 3.1 and Lemma 3.2 we have proved that in the case (i) the differential systems (1.2) either satisfy the conditions of Theorem 1.1, or have at most four functionally independent polynomial first integrals. Next we consider the case (ii).
In case (ii) Δ 1 /A 2 1 has the form m 2 1 n 2 1 /(n 1 + m 1 ) 2 with m, n ∈ Z \ {0} coprime. So it follows from the expressions of Δ 1 and A 1 that Δ 1 should be a square of (a 2 − a 1 Without loss of generality we can write it as (3.10) and it is easy to check that Then, from (3.10) and (3.11) equating to zero the coefficients of the monomials in the variables x 2 and y 2 we have Note that the first four conditions are inside the conditions of Theorem 1.1. Now we consider the last condition.
Lemma 3.3. The differential systems (1.2) under condition either satisfy the conditions of Theorem 1.1, or the eigenvalues of the singularity S 2 do not have a fifth linearly independent resonant lattice.
Proof. At the singularities S 2 , the 6-tuple of eigenvalues of the linear part of the differential systems (1.2) are given in (3.5). Direct calculations show that under the condition s 5 , the equation Δ 2 = 0 yields that either b 2 = b 3 = 0, which is not possible since otherwise b i = 0 for i = 1, 2, 3, or L 2 1 = 1 and then b 1 = b 3 , b 2 = a 1 − a 2 = a 3 − a 2 . Hence a 1 = a 3 . Thus, we are under the assumptions of Theorem 1.1. Then, under condition s 5 either Δ 2 = 0 and we are under the assumptions of Theorem 1.1, or Δ 2 = 0. Now, working in a similar way as we did for the singularities S 1 for studying if there is a fifth linearly independent resonant lattice at S 1 , we need to check if A 2 − √ B 2 / A 2 + √ B 2 = 0 is a rational number. For S 2 under the condition s 5 we have that )) 2 · Since L 1 = 0 this shows that there always exist infinitely many singularities S 2 which cannot satisfy condition (3.6). At these singularities S 2 the eigenvalues do not have a fifth linearly independent resonant lattice. rational number. Since on either s 6 , . . . , s 9 we have and since L 2 1 (K 2 1 − 4L 2 1 ) = 0 this shows that there always exist infinitely many singularities S 2 which cannot satisfy condition (3.6). At these singularities S 2 the eigenvalues do not have a fifth linearly independent resonant lattice.
Now direct calculations show that under one of the conditions s 10 , s 11 using that K 2 1 = 4L 2 1 , the equation Δ 2 = 0 yields b 3 = 0 which is not possible since otherwise b i = 0 for i = 1, 2, 3. Then, under one the conditions s 10 , s 11 we have Δ 2 = 0. Now working in a similar way as we did for the cases s 6 , . . . , s 9 , we must have that under one of the conditions s 10 , s 11 , condition (3.6) must hold. However, since Δ 2 /A 2 2 = N 1 N 2 /D 1 with and since L 1 K 1 (K 2 1 − 4L 2 1 ) = 0 this shows again that there always exist infinitely many singularities S 2 which cannot satisfy condition (3.6). At these singularities S 2 the eigenvalues do not have a fifth linearly independent resonant lattice.
From Theorem 3.1 and Lemma 3.4 we have proved that in the case (ii.2), the differential systems (1.2) either satisfy the conditions in Theorem 1.1 or have at most four functionally independent polynomial first integrals.
In short, if cases (i) or (ii) hold then the conditions given in Theorem 1.1 are satisfied and by Corollary 1.2 the differential systems (1.2) are completely integrable. If cases (i) and (ii) do not hold then by Theorem 3.1 the differential systems (1.2) can have at most four analytic integrals in a neighborhood of a point of S. Consequently the differential systems (1.2) have at most four functionally independent polynomial first integrals. This completes the proof of Theorem 1.3