Rich quasi-linear system for integrable geodesic flows on 2-torus

TL;DR

This paper links the integrability of geodesic flows on a 2-torus with the existence of solutions to a rich system of quasi-linear conservation laws, providing a new approach to understanding integrability conditions.

Contribution

It introduces a novel connection between integrability of geodesic flows and a rich quasi-linear system, framing the problem as a question of solutions to these equations.

Findings

The existence of polynomial first integrals corresponds to solutions of a rich quasi-linear system.

The integrability problem reduces to finding smooth (quasi-)periodic solutions of this system.

The system's structure offers new insights into the conditions for integrability on the 2-torus.

Abstract

Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Rich quasi-linear system.