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Infinitesimal symmetries of a partial differential equation (PDE) can bedefined algebraically as the solutions of the linearization (Frechetderivative) equation holding on the space of solutions to the PDE, and they arewell-known to comprise a linear space having the structure of a Liealgebra.Solutions of the adjoint linearization equation holding on the space ofsolutions to the PDE are called adjoint-symmetries. Their algebraic structurefor general PDE systems is studied herein. This is motivated by thecorrespondence between variational symmetries and conservation laws arisingfrom Noether's theorem, which has a well-known generalization tonon-variational PDEs, where infinitesimal symmetries are replaced byadjoint-symmetries, and variational symmetries are replaced by multipliers(adjoint-symmetries satisfying a certain Euler-Lagrange condition). Severalmain results are obtained. Symmetries are shown to have three different linearactions on the linear space of adjoint-symmetries. These linear actions areused to construct bilinear adjoint-symmetry brackets, one of which is apull-back of the symmetry commutator bracket and has the properties of a Liebracket. The brackets do not use or require the existence of any localvariational structure (Hamiltonian or Lagrangian) and thus apply to general PDEsystems. One of the symmetry actions is shown to encode a pre-sympletic(Noether) operator and a symplectic 2-form, which lead to the construction of asymplectic 2-form and Poisson bracket for evolution systems. The generalizedKdV equation in potential form is used to illustrate all of the results.

Symmetry actions and brackets for adjoint-symmetries. I: Main results and applications