An L-A pair for the Apel'rot system and a new integrable case for the Euler-Poisson equations on so(4)xso(4)

TL;DR

This paper introduces a new integrable case for the Euler-Poisson equations on so(4)xso(4) using an L-A pair, providing explicit integrals and challenging existing classification theorems.

Contribution

It presents a novel L-A pair for the Apel'rot case and generalizes it to discover a new integrable system in four dimensions.

Findings

Derived an L-A pair for the Apel'rot case.

Established a new integrable case for Euler-Poisson equations.

Provided explicit integrals in involution and corrected a classification theorem.

Abstract

We present an L-A pair for the Apel'rot case of a heavy rigid 3-dimensional body. Using it we give an algebro-geometric integration procedure. Generalizing this L-A pair, we obtain a new completely integrable case of the Euler-Poisson equations in dimension four. Explicit formulae for integrals which are in involution are given. This system is a counterexample to one well known Ratiu's theorem. Corrected version of this classification theorem is proved.