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There are 13 equivalence classes of 2D second order quantum and classicalsuperintegrable systems with nontrivial potential, each associated with aquadratic algebra of hidden symmetries. We study the finite and infiniteirreducible representations of the quantum quadratic algebras though theconstruction of models in which the symmetries act on spaces of functions of asingle complex variable via either differential operators or differenceoperators. In another paper we have already carried out parts of this analysisfor the generic nondegenerate superintegrable system on the complex 2-sphere.Here we carry it out for a degenerate superintegrable system on the 2-sphere.We point out the connection between our results and a position dependent massHamiltonian studied by Quesne. We also show how to derive simple models of theclassical quadratic algebras for superintegrable systems and then obtain thequantum models from the classical models, even though the classical and quantumquadratic algebras are distinct.

Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D