Foliation of the Phase Space for the Kepler Problem with Anisotropic Perturbations

We study a particular perturbation of the Kepler problem given by the potential U(r)=?1r?br2(1+cos2) , where b and ? are the perturbation parameters. This problem has two first integrals in involution: the first one is the well known Hamiltonian H=(pr2+p2r2)?1r?br2(1+cos2); the second one is given by G=p22?b(1+cos2). The sets H?1(h),G?1(g) and H?1(h)G?1(g) are invariant under the flow of the Hamiltonian system. From here we obtain a nice foliation of the phase space. In this paper we study the topology of the above foliation.