Andrzej J. Maciejewski, Maria Przybylska
Darboux Polynomials and First Integrals of Natural Polynomial Hamiltonian Systems 
(348K, postscript)

ABSTRACT.  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.