  The study of symmetries of partial differential equations (PDEs) has been
traditionally treated as a geometrical problem. Although geometrical methods
have been proven effective with regard to finding infinitesimal symmetry
transformations, they present certain conceptual difficulties in the case of
matrix-valued PDEs; for example, the usual differential-operator representation
of the symmetry-generating vector fields is not possible in this case. An
algebraic approach to the symmetry problem of PDEs is described, based on
abstract operators (characteristic derivatives) which admit a standard
differential-operator representation in the case of scalar-valued PDEs.
