The rolling of a body with a rotor on a moving supporting sphere

The rolling without slipping of a body with a rotor on a mobile supporting sphere in a uniform gravitational field is considered. The boundary of the body in the contact area with the support is part of a spherical surface. The central ellipsoid of inertia of the system (body + rotor) is an ellipsoid of revolution, the axis of which passes through the geometrical centre of the sphere, not, generally speaking, coinciding with the system mass centre. The supporting sphere is displaced translationally in an arbitrary way and is rotated around a vertical axis. The complete system of equations of motion of the supporting body and the rotor is obtained. Two integral equations of motion are obtained in the case of a solid of revolution. In the case when the body is a homogeneous sphere, four integral equations of motion are obtained, where the coordinates of the point of contact of the sphere and the supporting sphere are determined by quadratures, and all possible trajectories of the point of contact of the sphere and the body are indicated.