Integrable systems on the sphere associated with genus three algebraic curves

TL;DR

This paper introduces new separation variables for integrable systems on the sphere with higher-order integrals, detailing transformations, quadratures, and potential deformations to advance understanding of these complex systems.

Contribution

It provides explicit canonical transformations and quadratures for integrable systems on the sphere associated with genus three algebraic curves, a novel approach in this area.

Findings

Explicit separation variables for higher-order integrals

Canonical transformations between physical and separation variables

Discussion of integrable deformations of initial systems

Abstract

New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.