Completeness of multiseparable superintegrability in two dimensions

For complex Euclidean 2-space and the complex 2-sphere, we have found all classical and quantum superintegrable systems that a polynomial correspond to nondegenerate potentials. These potentials have the property that a polynomial associated with each of them is a quadratic algebra. Furthermore, each of these superintegrable systems admits separation of variables in more than one coordinate system. For degenerate superintegrable systems, both properties may be violated.