Invariants of homogeneous ordinary differential equations

In this paper we consider Y-homogeneous smooth vector fields. By means of a simple application of the Frobenius theorem, we prove that in a neighbourhood of a nonsingular point of a Y-homogeneous vector field there exist n − 1 independent and Y-homogeneous first integrals of it. We present a general method that can be applied to perform the reduction of a Y-homogeneous system of ordinary differential equations. We analyze the existence problem of densities of local invariant measures of Y-homogeneous systems. As applications of our approach, we demonstrate how to distinguish a class of three-dimensional systems with an explicitly given first integral, how to construct a Poisson structure for a system possessing a first integral and a compatible vector field, and how to obtain a global first integral for certain three-dimensional divergence-free systems.