Higher order superintegrability of separable potentials with a new approach to the Post-Winternitz system

The higher order superintegrability of separable potentials is studied. It is proved that these potentials possess (in addition to the two quadratic integrals) a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions. Some systems related to the harmonic oscillator, such as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related to the Kepler problem. In this way, a new proof of the superintegrability of the Post-Winternitz system is presented and the explicit expression for the integral is obtained. Finally, the relations between the superintegrable systems with quadratic constants of motion (separable in several different coordinate systems) and the superintegrable systems with higher order constants of motion are analyzed.