On Mikhailov's reduction group

• We propose and consider in a systematic way a generalization of reduction group.
• Our considerations let us apply the inverse scattering method to classes of nonlocally reduced equations.
• Several examples of nonlocal reductions to illustrate the general ideas have been considered in more detail.

We propose a generalization of the notion of reduction group which provides group-theoretical tools to study in a uniform way certain classes of nonlocal S-integrable equations like Ablowitz–Musslimani's nonlocal Schrödinger equation. Another benefit of the generalization to be presented here is that it supplies us with a systematic approach to construct solutions to S-integrable equations with prescribed symmetries.