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Harmonic oscillator and the Kepler problem are superintegrable systems whichadmit more integrals of motion than degrees of freedom and all these integralsare polynomials in momenta. We present superintegrable deformations of theoscillator and the Kepler problem with algebraic and rational first integrals.Also, we discuss a family of superintegrable metrics on the two-dimensionalsphere, which have similar first integrals.

Elliptic curve arithmetic and superintegrable systems