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A classical (or quantum) second order superintegrable system is an integrablen-dimensional Hamiltonian system with potential that admits 2n-1 functionallyindependent second order constants of the motion polynomial in the momenta, themaximum possible. Such systems have remarkable properties: multi-integrabilityand multi-separability, an algebra of higher order symmetries whoserepresentation theory yields spectral information about the Schroedingeroperator, deep connections with special functions and with QES systems. Here weannounce a complete classification of nondegenerate (i.e., 4-parameter)potentials for complex Euclidean 3-space. We characterize the possiblesuperintegrable systems as points on an algebraic variety in 10 variablessubject to six quadratic polynomial constraints. The Euclidean group acts onthe variety such that two points determine the same superintegrable system ifand only if they lie on the same leaf of the foliation. There are exactly 10nondegenerate potentials.

Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties