Bounded motion in a two-body system consisting of a solid body and a material point

In this paper we prove the existence of bounded motions for an isolated system consisting of a solid bodyB ₁ and a material pointB ₂ moving under their mutual gravitational attraction. We also consider the special case where the mass ofB ₁ is symmetrically distributed with respect to three mutually perpendicular planes passing through its mass center andB ₂ moves on one of these planes. We study the types of the regions of possible motion and the ways of their evolution as the energy or the angular momentum of the system changes. As an example we present some results from a numerical study of the case whereB ₁ is a homogeneous prolate spheroid.