Classical and Quantum Superintegrability of Stackel Systems

In this paper we discuss maximal superintegrability of both classical and quantum Stackel systems. We prove a sufficient condition for a flat or constant curvature Stackel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stackel transform to preserve maximal superintegrability and we apply this condition to our class of Stackel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.