Recursion and group structures of soliton equations

The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered.