On additional integrals of motion in classical and quantum mechanics

For the particle motion in the field of a central potential an analysis of the situation with the existence of additional integrals of motion in classical mechanics is given. It is shown that all existing single-valued integrals of motion (including the Runge-Lenz vector for the Coulomb potential and oscillator integrals) can be deduced from the equations of orbit and trajectory. Also, from these equations additional single-valued integrals for an arbitrary central potential may be obtained. Also the reasons for the difference of the symmetry group in quantum and classical mechanics are discussed