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Second-order superintegrable systems in dimensions two and three areessentially classified. With increasing dimension, however, the non-linearpartial differential equations employed in current methods become unmanageable.Here we propose a new, algebraic-geometric approach to the classificationproblem - based on a proof that the classification space for irreduciblenon-degenerate second-order superintegrable systems is naturally endowed withthe structure of a quasi-projective variety with a linear isometry action. Onconstant curvature manifolds our approach leads to a single, simple andexplicit algebraic equation defining the variety classifying superintegrableHamiltonians that satisfy all relevant integrability conditions generically. Inparticular, this includes all non-degenerate superintegrable systems known todate and shows that our approach is manageable in arbitrary dimension. Our workestablishes the foundations for a complete classification of second-ordersuperintegrable systems in arbitrary dimension, derived from the geometry ofthe classification space, with many potential applications to relatedstructures such as quadratic symmetry algebras and special functions.

An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension