Hidden symmetries mapped for gravity coupled to nonlinear electrodynamics

Stationary, axisymmetric gravity coupled to scalar and electromagnetic fields admits a rich structure of symmetries that can generate new exact solutions from known ones. This work catalogs the full set of visible, hidden, sectorial, and discrete symmetries for both standard Einstein-Maxwell-Scalar and the nonlinear ModMax variant, showing how Ehlers and Harrison transformations act on distinct sectors of the potential space. A key finding is that requiring electric and magnetic Harrison symmetries to coexist forces the ModMax theory into a frozen regime, selecting a specific sector of the nonlinear theory. The framework is applied to produce charged and rotating solution branches from simple vacuum Weyl seeds.