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.