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We prove that the set of non-degenerate second order maximallysuperintegrable systems in the complex Euclidean plane carries a naturalstructure of a projective variety, equipped with a linear isometry groupaction. This is done by deriving the corresponding system of homogeneousalgebraic equations. We then solve these equations explicitly and give adetailed analysis of the algebraic geometric structure of the correspondingprojective variety. This naturally associates a unique planar line triplearrangement to every superintegrable system, providing a geometric realisationof this variety and an intrinsic labelling scheme. In particular, our resultsconfirm the known classification by independent, purely algebraic means.

An algebraic geometric classification of superintegrable systems in the Euclidean plane