Elliptic curve arithmetic and superintegrable systems

TL;DR

This paper explores superintegrable systems, presenting deformations of classical problems like the harmonic oscillator and Kepler problem, with new algebraic and rational integrals, and introduces superintegrable metrics on the sphere.

Contribution

It introduces superintegrable deformations of classical systems with novel algebraic and rational integrals, expanding the understanding of superintegrability.

Findings

Deformations of oscillator and Kepler problems with new integrals

Existence of superintegrable metrics on the sphere

Identification of algebraic and rational first integrals

Abstract

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