  A method to construct Hamiltonian theories for systems of both ordinary and
partial differential equations is presented. The knowledge of a Lagrangian is
not at all necessary to achieve the result. The only ingredients required for
the construction are one solution of the symmetry (perturbation) equation and
one constant of the motion of the original system. It turns out that the
Poisson bracket structure for the dynamical variables is far from being
uniquely determined by the differential equations of motion. Examples in
classical mechanics as well as in field theory are presented.
