Involution analysis of the partial differential equations characterizing Hamiltonian vector fields

In a recent article, certain underdetermined linear systems of partial differential equations connected with Lie-Poisson structures have been studied. They were constructed via power series solutions of the evolution equation for a given Hamiltonian. We extend the results to arbitrary Poisson manifolds, correct an error in the case of degenerate Poisson structures, and show that these linear systems simply characterize Hamiltonian vector fields. Our basic tool is the formal theory of differential equations with its central concept of an involutive system. (C) 2003 American Institute of Physics.