On the first integrals of equation of motion of a heavy rotational symmetric body on a perfectly rough plane

The problem on a motion of a heavy rigid dynamically symmetric body, bounded by a surface of rotation, on a fixed horizontal plane without sliding is considered. In a case, when the bounded surface satisfies to certain condition, the explicit form of two first integrals of equation of motion of a body, in addition to the energy integral is found. These integrals are linear with respect to generalized velocities. This work supplements and develops results, received in classical work.