Rational integrals of 2-dimensional geodesic flows: new examples

TL;DR

This paper constructs explicit examples of 2D Riemannian metrics with geodesic flows that admit rational first integrals, expanding the known class of superintegrable systems with both polynomial and rational integrals.

Contribution

It introduces new explicit metrics and integrals for 2D geodesic flows, demonstrating the existence of superintegrable systems with rational integrals.

Findings

Explicit examples of metrics with rational integrals are provided.

New superintegrable systems with polynomial and rational integrals are identified.

The paper advances understanding of integrability in 2D geodesic flows.

Abstract

This paper is devoted to searching for Riemannian metrics on 2-surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.