  Symmetries and adjoint-symmetries are two fundamental (coordinate-free)
structures of PDE systems. Recent work has developed several new algebraic
aspects of adjoint-symmetries: three fundamental actions of symmetries on
adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the
range of a symmetry action; a generalized Noether (pre-symplectic) operator
constructed from any non-variational adjoint-symmetry. These results are
illustrated here by considering five examples of physically interesting
nonlinear PDE systems -- nonlinear reaction-diffusion equations, Navier-Stokes
equations for compressible viscous fluid flow, surface-gravity water wave
equations, coupled solitary wave equations, and a nonlinear acoustic equation.
