On gyrostat motions characterized by linear in angular velocity invariant relations

The paper concerns a motion of a non-autonomous gyrostat with fixed point in the gravity field under the assumption that the direction of variable gyrostatic momentum is fixed in the rotating frame. For the balanced gyrostat it is shown that motions characterized by two linear invariant relations represent rotations about a permanent axis. The exact solutions of the motion equations are obtained for a heavy gyrostat in the case of one or two linear in angular velocity invariant relations.