On a class of integrable systems with a quartic first integral

TL;DR

This paper extends the classification of integrable geodesic flows on 2D manifolds with a quartic first integral, identifying new models on S^2, H^2, and R^2, including known systems like Kovalevskaya's.

Contribution

It generalizes previous results by explicitly integrating the differential system for the quartic integral and describing new global models on classical surfaces.

Findings

Derived a finite-parameter family of integrable models

Identified models on S^2, H^2, R^2

Recovered Kovalevskaya's system and its generalization

Abstract

We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some details and leads to a class of models living on the manifolds S^2, H^2 or R^2. As special cases we recover Kovalevskaya's integrable system and a generalization of it due to Goryachev.