**Next:** Brief
Summary of Yoshida's **Up:** NON-INTEGRABILITY
OF THE MOTION **Previous:** NON-INTEGRABILITY
OF THE MOTION

More than one century ago, the theory of orbits went through an important qualitative change when Bruns and then Poincaré (see [10]) proved by means of several methods and hypotheses that the three body problem has no more first integrals apart from the ten classical ones. Since then, great progress has been made in the knowledge of the behaviour of such non-integrable systems (including chaos), as well as in the methods which allow us to investigate the non-integrability. A panorama of the evolution of the ideas and results concerning the study of problems of several bodies can be found in the usual treatises - for instance, see [9] or [5].

However, questions about non-integrability of problems of satellites
of external bodies remained open till this decade. Irigoyen and Simó
[3] published the first result along this
line, proving that the main problem of the satellite of an oblate primary,
or *J*_{2}-problem, was not completely integrable through
meromorphic integrals. The proof was based on some theorems by Ziglin [14]
and Yoshida [11], which will be commented
on briefly in section 2. Later, an analogous method allowed Ferrándiz
and Sansaturio [1] to establish that the
*J*_{22}-problem did not have a complete system of first integrals.
Such integrals were supposed to be rational since, in three dimensions,
the non-integrability according to Ziglin depends on a condition of rational
independence of the eigenvalues of certain matrices which does not allow
us to use the continuity arguments as happens in dimension 2.

In [2], the present authors set up the
non-integrability through meromorphic integrals of the truncation of the
zonal satellite problem at any order. Then in [6],
we solved the case in which the perturbation was an arbitrary sectorial
harmonic and established that the main satellite problem, or (*J*_{2}+*J*_{22})-problem,
is not completely integrable either.

In this paper we gather together the above mentioned results, either with much briefer proofs or by generalizing them.