# The Kepler problem: polynomial algebra of non-polynomial first integrals

## TL;DR

This paper explores the algebraic structure of non-polynomial first integrals in superintegrable systems related to elliptic curves, revealing new insights into their polynomial algebraic properties.

## Contribution

It introduces the polynomial algebra of non-polynomial first integrals linked to elliptic curve divisors, expanding understanding of symmetries in superintegrable systems.

## Key findings

- Polynomial algebra of non-polynomial first integrals is characterized.
- Connection between elliptic integrals and orbit determination is established.
- New algebraic structures for fixed points on elliptic curves are identified.

## Abstract

The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. Algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas algebra of the first integrals associated with coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of non-polynomial first integrals of superintegrable systems associated with elliptic curves.