Superintegrable metrics on surfaces admitting integrals of degrees 1 and 4

TL;DR

This paper classifies and analyzes Riemannian metrics on surfaces with superintegrable geodesic flows featuring integrals linear and quartic in momenta, providing local descriptions, solutions, and symmetry insights.

Contribution

It offers a local differential equation-based classification and explicit solutions for such superintegrable metrics, expanding understanding of their algebraic and geometric properties.

Findings

Local differential equations characterize the metrics

Explicit solutions for certain metrics on the sphere

Analysis of the symmetry group of the problem

Abstract

We study Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral quartic in momenta. The main results of the work are local description of such metrics in terms of ordinary differential equations, integration of the equations, and description of the corresponding Poisson algebra of integrals of motion. We also give examples of such metrics that can be extended to the sphere $S^2$, and study the group of symmetries of the problem.