We consider an autonomous polynomial Hamiltonian differential system of n degrees of freedom. In this Letter we study relationships between the existence of Darboux polynomials and additional Darboux polynomials for polynomial Hamiltonian systems. Also we extend some ideas of the Darboux theory of integrability for polynomial differential systems to general polynomial Hamiltonian systems taking into account the multiplicity of Darboux polynomials.
Darboux polynomials and first integrals of polynomial Hamiltonian systems