Constants of motion and the variational equations

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 deltau which satisfies the linear equation is of the form deltau = [u,T], then T is a constant of motion. Further, partialT/partialt is also a constant. The sequence [T,partialT/partialt], [[T,partialT/partialt],T],. . . is shown to produce other nontrivial constants