New families of conservative systems on $S^2$ possessing an integral of fourth degree in momenta

TL;DR

This paper introduces new families of integrable conservative systems on the sphere $S^2$ that have a fourth-degree integral in momenta, expanding known examples like Kovalevskaya and Goryachev.

Contribution

The paper presents novel examples of conservative systems on $S^2$ with a fourth-degree integral, extending previous known families including Kovalevskaya and Goryachev's systems.

Findings

New families of integrable systems on $S^2$ with fourth-degree integrals.

Extension of Goryachev's family of systems.

Broader understanding of integrable conservative dynamics on spheres.

Abstract

There is a well-known example of integrable conservative system on $S^2$, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. Goryachev proposed a one-parameter family of examples of conservative systems on $S^2$ possessing an integral of fourth degree in momenta which includes the case of Kovalevskaya. In this paper we proposed new examples of conservative systems on $S^2$ possessing an integral of fourth degree in momenta.