  We consider systems of ordinary differential equations with quadratic
homogeneous right hand side. We give a new simple proof of a result already
obtained in [8,10] which gives the necessary conditions for the existence of
polynomial first integrals. The necessary conditions for the existence of a
polynomial symmetry field are given. It is proved that an arbitrary homogeneous
first integral of a given degree is a linear combination of a fixed set of
polynomials.
