Rational integrals of quasi-homogeneous dynamical systems

Dynamical systems, described by quasi-homogeneous systems of differential equations with polynomial right-hand sides, are considered. The Euler–Poisson equations from solid-state dynamics, as well as the Euler–Poincaré equations in Lie algebras, which describe the dynamics of systems in Lie groups with a left-invariant kinetic energy, can be pointed out as examples. The conditions for the existence of rational first integrals of quasi-homogeneous systems are found. They include the conditions for the existence of invariant algebraic manifolds. Examples of systems with rational integrals which do not admit of first integrals that are polynomial with respect to the momenta are presented. Results of a general nature are also demonstrated in the example of a Hess–Appel’rot invariant manifold from the dynamics of an asymmetric heavy top.