Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are
considered. We present all real standard potentials that allow separation of
variables in polar coordinates and admit an independent fourth-order integral
of motion. The general form of the potentials satisfies a linear ODE. In the
classical case, the standard potentials coincide with the
Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models. In the
quantum case new superintegrable systems are obtained, in addition to the TTW
and PW ones. Their classical limit is free motion.Comment: 30 page
