Lax representations and variational Poisson structures for magnetohydrodynamics equations

We find two Lax representations for the reduced magnetohydrodynamics equations (rmhd) and construct a local variational Poisson structure (a Hamiltonian operator) for them. Its inverse defines a nonlocal symplectic structure for the same equations. We describe the action of both operators on the second-order cosymmetries and on the infinitesimal contact symmetries of rmhd, respectively. The reduction of rmhd by the symmetry of shifts along the z-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics (imhd). Applied to the Lax representations and the variational Poisson structure of rmhd, the reduction provides analogous constructions for imhd.