# Discretization and superintegrability all rolled into one

## TL;DR

This paper explores a novel approach linking Abelian integrals to physical processes by interpreting fixed points as parameters for discretization or as integrals of motion, potentially enhancing understanding of integrable systems.

## Contribution

It introduces a new interpretation of Abelian integrals in physics, connecting them to discretization parameters and first integrals, expanding their application scope.

## Key findings

- Proposes a new interpretation of Abelian integrals in physical systems.
- Links fixed points in Abelian integrals to discretization and integrals of motion.
- Suggests potential applications in analyzing integrable systems.

## Abstract

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.