2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58455We 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.14 pagesMathematical PhysicsSymplectic Geometry58F05An L-A pair for the Apel'rot system and a new integrable case for the Euler-Poisson equations on so(4)xso(4)text