Generalised Harrison transformations and black diholes in Einstein-ModMax

A bstract Einstein-Maxwell theory has powerful solution generating techniques which include Harrison transformations in the Ernst formalism. We construct generalised Harrison transformations that preserve the purely magnetic or purely electric sector in Einstein-ModMax (EMM) theory. Thus, they serve as solution generating techniques within these sectors for this model of non-linear electrodynamics minimally coupled to gravity. As an application we rederive several known exact solutions of EMM and a new solution, black diholes, describing two extremal BHs in equilibrium, with opposite magnetic charges, whose attraction is balanced by their embedding in the Melvin magnetic universe of this model. As a further generalisation, we consider Einstein-dilaton-ModMax theory, and provide the extremal charged BHs and black diholes also in this model.