The Euler–Poisson equations; an elementary approach to partial integrability conditions, Goryachev–Chaplygin and beyond

We consider, in the complex domain, the Euler–Poisson equations describing the motion of a heavy rigid body about a fixed point. The real domain is a particular case of it. The Euler–Poisson equations admit three functionally independent first integrals H1, H2, H3, i.e. the area, geometrical and energy first integrals. In four cases (Euler, Lagrange, Kovalevskaya, kinetic symmetry) a fourth functionally independent first integral appears. It can be found among polynomials that do not depend on all variables. We study when, apart from the four cases above, the Euler–Poisson equations, restricted to the level manifolds of H1, H2 and H3 and all their mutual intersections, admit a new first integral which does not depend on all variables. In this way we cover the partially integrable Goryachev–Chaplygin case and describe, in the complex domain, a new class of partially integrable cases on the level manifold {H1=0, H2=0}. We also deduce their uniqueness. We also cover the Sretenskii case of partial integrability of the gyrostat equations (which generalizes the Goryachev–Chaplygin case) and describe a new class of their integrable cases in the complex domain. We provide a general quasi-algorithmic method to find all these cases and corresponding partial integrals. The use of computer algebra is unavoidable to carry out our investigations. By following the link (https://sdrive.cnrs.fr/public.php/dav/files/bKmGokMQnM5Jo5f/?accept=zip), the reader can verify all reported computations using EPEPcomp.zip available at that link. Note that a public user can only download files without the edition option.