Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.