Potential Space Symmetries in Ernst-like Formulations of Einstein-Maxwell/ModMax-Scalar field Theories

We complete the visible, hidden, sectorial, and discrete symmetries of Ernst-like potential spaces in stationary, axisymmetric Einstein-Maxwell-Scalar Field (EMSF) and Einstein-ModMax-Scalar Field (EMMSF) theories. In the real potential space $(f,\epsilon,\psi,\chi,\kappa)$, we determine the exact visible symmetries and their solvable Lie algebra. We characterize the hidden symmetries on invariant subspaces: Ehlers acts in the gravito-rotational sector, while electric and magnetic Harrison transformations act in static electromagnetic sectors. In the frozen EMMSF regime, $v=v_0,\ w=w_0$, we show how EMSF sectorial transformations are deformed in ModMax theory. We also show that coexistence of electric and magnetic sectorial Harrison transformations imposes $d w=0$ and $d[(v^2+w^2)/w]=0$, selecting precisely the frozen ModMax sector. We study the Hamiltonian formulation, Noether charges, and Casimir invariants of the sectorial algebras. In harmonic branches of the $A,B,C$ one-forms, the affine geodesic energy is constant, so the quadrature for $k$ is controlled by the affine-geodesic Hamiltonian. The functions $\omega$ and $A_\varphi$ follow from Noether charges along the Killing directions of $\epsilon$ and $\chi$, and are written using dual harmonic functions. We examine the electric and magnetic Lewis-Weyl-Papapetrou frames and their discrete map, which sends $\kappa\mapsto\kappa^{-1}$, $\psi\mapsto\chi$, $\chi\mapsto-\psi$, and $\epsilon\mapsto\epsilon-\psi\chi$. Finally, we apply the sectorial transformations to harmonic scalar--acuum Weyl seeds with independent gravitational and scalar harmonics. Frozen-ModMax Harrison maps generate charged branches, while Ehlers generates the gravito-rotational branch. For these solutions we give the final quadratures for $k$, $\omega$, and $A_\varphi$.