  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 with the harmonic oscillator, as the generalized SW system
and the TTW system, were studied in previous papers; now a similar analysis is
presented for superintegrable systems related with the Kepler problem. In this
way, a new proof of the superintegrability of the Post-Winternitz system is
presented and the explicit expression of the integral is obtained. Finally, the
relations between the superintegrable systems with quadratic constants of the
motion (separable in several different coordinate systems) and the
superintegrable systems with higher-order constants of the motion are analyzed.
