General Nth-order superintegrable systems separating in polar coordinates

The general description of superintegrable systems with one polynomial integral of order $N$ in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean plane. We consider Hamiltonian systems allowing separation of variables in polar coordinates. The classical and the quantum cases are treated. A new infinite family of superintegrable potentials in terms of the sixth Painlev\'e transcendent $P_6$ is presented.