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

The problem of the motion of a dynamically symmetric gyrostat without slipping on a fixed horizontal plane is investigated. When the surface of the gyrostat and the distribution of the masses in it satisfy a certain condition, supplementing and developing the results obtained by Mushtari [Mushtari KhM. The rolling of a heavy solid of revolution on a fixed horizontal plane. Mat Sbornik 1932; 39(1, 2):105–26], an explicit form of two first integrals of the equations of motion of the gyrostat, in addition to the energy integral, is presented.