On Local Description of Two-Dimensional Geodesic Flows with a Polynomial First Integral

TL;DR

This paper constructs multiparametric families of 2D metrics with polynomial first integrals, linking integrable geodesic flows to semi-Hamiltonian systems and providing explicit solutions via the generalized hodograph method.

Contribution

It introduces a new approach to describe integrable 2D geodesic flows using semi-Hamiltonian systems and constructs explicit metric families with polynomial integrals.

Findings

Infinite conservation laws identified

Existence of commuting flows established

Explicit metric solutions obtained via generalized hodograph method

Abstract

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.