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Discretization and superintegrability all rolled into one

Abelian integrals arise in the mathematical description of various physical processes. According to Abel's theorem these integrals are related to motion of a set of points along a plane curve around fixed points, which are relatively little used in physics applications. We propose to interpret coordinates of the fixed points either as parameters of exact discretization or as additional first integrals for equations of motion reduced to Abelian quadratures.