The Euler-Poisson equations: An elementary approach to integrability conditions

Abstract

We consider the Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point with parameters in a complex domain. We suppose that these equations admit a first integral functionally independent of the three already known integrals which does not depend on all the variables. We prove that this may happen only in the already known three integrable cases or in the trivial case of kinetic symmetry. We provide a method for finding such a fourth integral, when it exists.