New integrable problems of motion of a rigid body with a particle oscillating or bouncing in it (Q1371264)

We introduce the general integrable case in which a particle moves on the axis of dynamical symmetry of a rigid body and is subject to an arbitrary conservative force that depends only on the distance from the fixed point. Separation of variables is accomplished and the solution is reduced to quadratures. As a special version of this problem, the case when the particle bounces elastically between two points is discussed.