We focus on partial Hamiltonian systems for the characterization of their operators and related first integrals. Firstly, it is shown that if an operator is a partial Hamiltonian operator which yields a first integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a first integral. Thirdly, characterization of partial Hamiltonian operators and related first integral conditions are provided for the partial Hamiltonian system. Applications to mechanics are presented to illustrate the theory.
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