Introduction

The search for the integrability of a given problem is a classical issue. Although non-integrability may be expected as a generic property, its proof is usually hard to obtain even for simple cases and, in the last decades, a lot of mathematical tools have been developed towards this end (Ziglin [14]; Yoshida [8], [9], [10], [11], [12]). The three and n-body problems are those most profusely treated in the Celestial Mechanics literature. However, in recent years, some attention has been paid to the satellite problem, i.e., the orbital motion of a point mass around a planet of arbitrary shape.

In 1993, Irigoyen and Simó [3] succeeded in proving the non-integrability of the main problem of the satellite of an oblate primary, or J₂-problem, and this was a highly expected result in the scientific community. Basically, they applied a Theorem by Yoshida [10], that was based on Ziglin's theorem [14] for special Hamiltonian systems with a homogeneous potential, and concluded that there was no additional global meromorphic first integral besides the Hamiltonian itself.

More recently, the authors started to consider more general cases of the satellite problem. In this paper we report on the partial results obtained so far ([1]; [2]), that concern the J₂₂-problem, the general zonal problem and that with an arbitrary tesseral perturbation. Although these problems are not homogeneous either with respect to the properties required for the first integrals or with respect to the concrete approach and theorems applied in each case, the basic method is always Ziglin's theorem.

In section 2 we consider the problem of the zonal satellite. The aforementioned result about the non integrability of the J₂-problem is generalized to a perturbation made up of an arbitrary number of zonal harmonics. In this way, the non-integrability through meromorphic integrals of the zonal satellite problem truncated at any order is shown. It is worthwhile noticing that when the number of harmonics is infinite, there could be complete integrability, as occurs in Vinti's problem (Vinti, [6], [7]). Such a problem, closely related to Euler's problem of two-fixed centres of force (Szebehely, [5]), provides an example of a non trivial integrable Hamiltonian for which all the truncations of order n>2 (of the development of the potential in spherical harmonics) are non-integrable in the Liouville sense. A similar case appears in the Toda lattice, as was shown by Yoshida [11], and the non-integrability of truncations is expected to be a rather generic property, although the number of examples is limited. The inverse problem looks more interesting since, up to now, the existence of other infinite series of the development in zonal harmonics giving rise to complete integrability, apart from those of Vinti or the trivial one (Kepler's problem), is not known. In this respect, the heavy rigid body problem seems to have given rise to more integrable subcases (Ziglin, [13]).

In section 3 we present a further elaboration of our former result on the J₂₂-problem, that is, the problem of a satellite only perturbed by the main sectorial harmonic, for which we establish the non-existence of additional global rational integrals essentially by applying a theorem due to Yoshida [12] to an auxiliary problem. In spite of the fact that finding a body with the required moments of inertia and providing an example of such a problem is not a difficult task, this case can be mainly considered as a problem of mathematical interest. Nevertheless, it can also be considered as the first step in the treatment of those cases where the perturbation includes terms beyond the zonal ones and for which no component of the angular momentum is kept, so that no simplification is possible by means of a reduction in the number of degrees of freedom.