Dynamics of a point in the axisymmetric gravitational potential of a massive fixed ring and center

We consider the problem of three-dimensional motion of a passively gravitating point in the potential created by a homogeneous thin fixed ring and a massive point located in the center of the ring. The motion of a passively gravitating point admits two first integrals. We first consider the integrable case of an invariant motion in the equatorial plane and then consider the general case of three-dimensional motion, where we classify the possible trajectories of a point depending on the values of the first integrals. Finally, some previous results for similar problems are compared.