Generalized conditional symmetries and pre-Hamiltonian operators

In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand–Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs.