We find two Lax representations for the reduced magnetohydrodynamics equations ({ \sc rmhd}) and derive a B{ \"a}cklund transformation between the tangent and the cotangent coverings of these equations. Then, we study the action of the B{ \"a}cklund transformation on the second-order cosymmetries and the action of its inverse on the Lie symmetries of { \sc rmhd}. The action of the inverse transformation produces another Lax representation for { \sc rmhd}. The reduction of { \sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({ \sc imhd}). Applied to the Lax representations and the B{ \"a}cklund transformation of { \sc rmhd}, the reduction provides analogous constructions for { \sc imhd}. The action of the inverse B{ \"a}cklund transformation on the Lie symmetries of { \sc imhd} is expressed in terms of a new four-parameter Lax representation of these equations.
Categories: nlin.SI