Superintegrable Systems with Algebraic and Rational Integrals of Motion

Abstract

We consider superintegrable deformations of the Kepler problem and the harmonic oscillator on the plane and also superintegrable metrics on a two-dimensional sphere, for which the additional integral of motion is either an algebraic or a rational function of momenta. According to Euler, these integrals of motion take the simplest form in terms of affine coordinates of elliptic curve divisors.