Two-dimensional superintegrable metrics with one linear and one cubic integral

TL;DR

This paper classifies all local Riemannian metrics on surfaces with superintegrable geodesic flows having one linear and one cubic integral, providing new examples on the sphere and nonconstant curvature surfaces.

Contribution

It completely characterizes such metrics and introduces the first superintegrable metrics of nonconstant curvature on closed surfaces.

Findings

Identified all local metrics with the specified superintegrability properties.

Extended some metrics to the 2-sphere, creating new Hamiltonian systems.

Provided the first examples of superintegrable metrics with nonconstant curvature on closed surfaces.

Abstract

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and the first examples of superintegrable metrics of nonconstant curvature on a closed surface