On the polynomial integrability of a system motivated by the Riemann ellipsoid problem

Abstract

Abstract. We consider differential systems obtained by coupling two Euler-Poinsot systems. The motivation to consider such systems can be traced back to the Riemann ellipsoid problem. We provide new cases for which these systems are completely integrable. We also prove that these systems either are completely integrable or have at most four functionally independent polynomial first integrals.