  A number of characteristics of integrable nonlinear partial differential
equations (PDE's) for classical fields are reviewed, such as Backlund
transformations, Lax pairs, and infinite sequences of conservation laws. An
algebraic approach to the symmetry problem of PDE's is described, suitable for
PDE's the solutions of which are non-scalar fields (e.g., are in the form of
matrices). A method is proposed which, in certain cases, may "unify" the
symmetry and integrability properties of a nonlinear PDE. Application is made
to the self-dual Yang-Mills equation and the Ernst equation of General
Relativity.
