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

Abstract

We present an L-A pair for the Hess-Apel'rot case of a heavy rigid three-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 that are in involution are given. This system is a counterexample to one of Ratiu's theorems. A corrected version of this classification theorem is proved.