On the first integrals of equations of motion of a symmetric gyrostat on a perfectly rough plane

The problem on a motion of a heavy dynamically symmetric gyrostat, bounded by a surface of rotation, on a fixed horizontal plane without sliding is studied. In a case, when the bounded surface and distribution of mass of a gyrostat satisfy to certain condition, the explicit forms of two first integrals of equations of motion of a gyrostat, in addition to the energy integral, are found. These integrals are linear with respect to generalized velocities.