Hamiltonians separable in Cartesian coordinates and third-order integrals of motion

We present in this article all Hamiltonian systems in E(2) that are separable in Cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-de Vries equation and the Painleve transcendents. (C) 2004 American Institute of Physics.