  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
Painlev\'e transcendents.
