For field equations of Hamiltonian form the relation between constants of motion and solutions of the linearized equation is discussed. A known result is that the Poisson bracket of a constant of motion with the field variable solves the linearized equation. Here the following converse result is obtained: If a \ensuremath{\delta}u which satisfies the linear equation is of the form \ensuremath{\delta}u=[u,T], then T is a constant of motion. Further, \ensuremath{\partial}T/\ensuremath{\partial}t is also a constant. The sequence [T,\ensuremath{\partial}T/\ensuremath{\partial}t], [[T,\ensuremath{\partial}T/\ensuremath{\partial}t],T],. . . is shown to produce other nontrivial constants.