Canonical transformations of the extended phase space, Toda lattices and the Stackel family of integrable systems

We consider some examples of canonical transformations of the extended phase space, which map a completely integrable system into another completely integrable system. The proposed transformations are closed to the known Maupertuis-Jacobi mappings and to the Kepler change of the time. Using the Kepler transformation we construct new integrable systems related to the Toda lattices and the Stackel family of integrable systems.