Constructing conditionally integrable evolution systems in (1 + 1) dimensions: a generalization of invariant modules approach

Given a generalized (Lie–Bäcklund) vector field satisfying certain nondegeneracy assumptions, we explicitly describe all (1 + 1)-dimensional evolution systems that admit this vector field as a generalized conditional symmetry. The connection with the theory of symmetries of systems of ODEs and with the theory of invariant modules is discussed.