Regular precessions of a gyrostat with fixed point in the gravity field

This article represents an investigation of regular precessions of a gyrostat about the fixed inclined axis. For a gyrostat with variable gyrostatic momentum, whose direction is fixed in the rotating frame, necessary and sufficient conditions of regular precession existence are obtained. The inclination angle of precession axis dependence on both gyrostat parameters and the proper rotation axis direction is computed. The respective value of gyrostatic momentum and solutions of the motion equations are explicitly found. Among them the exact solution, which becomes the Grioli one in the case of zero gyrostatic momentum, is presented. New solution, which correspondents to the regular precession of a gyrostat when the precession velocity is twice as much than proper rotation one, is also obtained.