A locally integrable non-separable analytic geodesic flow

TL;DR

This paper constructs an explicit example of an analytic metric on a 2-torus that is non-separable yet locally integrable on an energy surface, using a KAM-like approach to control the dynamics.

Contribution

It provides a novel explicit example of a non-separable analytic metric with local integrability, advancing understanding of integrable systems on tori.

Findings

Explicit example of a non-separable analytic metric on T^2

The metric is locally integrable on an energy surface

Uses a KAM-like approach for construction

Abstract

We explicitely construct an example of an analytic metric on $T^2$ which is non-separable but it is locally integrable on an energy surface. The construction is based on a KAM-like approach and a careful control on what happens on the energy surface.