2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94804We 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 Painlevé transcendents.19 pages, Will be published in J. Math. PhysMathematical PhysicsDynamical Systemstext