Completeness of multiseparable superintegrability in E-2,(C)

The possibility that Schrodinger's equation with a given potential can sepa rate in more than one coordinate system is intimately connected with the no tion of superintegrability. Examples of this type of system are well known. In this paper we demonstrate how to establish a complete list of such pote ntials using essentially algebraic means. Our approach is to classify all n ondegenerate potentials that admit a pair of second-order constants of moti on. Here 'nondegenerate' means that the potentials depend on four independe nt parameters. This is carried out for two-dimensional complex Euclidean sp ace, though the method generalizes to other spaces and dimensions. We show that all these superintegrable systems correspond to quadratic algebras, an d we work out the detailed structure relations and their quantum analogues.