Darboux Polynomials and First Integrals of Natural Polynomial   Hamiltonian Systems
  We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals. 
Andrzej J. Maciejewski and Maria Przybylska