Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method

TL;DR

This paper investigates Riemannian metrics on 2-surfaces with integrable geodesic flows possessing high-degree polynomial first integrals, utilizing the generalized hodograph method to construct explicit examples.

Contribution

It introduces a systematic approach to find explicit solutions for high-degree polynomial integrals in geodesic flows on 2-surfaces using semi-Hamiltonian PDE systems.

Findings

Constructed explicit examples with polynomial degrees 3, 4, 5

Reduced the problem to semi-Hamiltonian PDE systems

Applied generalized hodograph method successfully

Abstract

We study Riemannian metrics on 2-surfaces with integrable geodesic flows such that an additional first integral is high-degree polynomial in momenta. This problem reduces to searching for solutions to certain quasi-linear systems of PDEs which turn out to be semi-Hamiltonian. We construct plenty of local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5. Our construction is essentially based on applying the generalized hodograph method.