Families of exact solutions of a 2D gravity model minimally coupled to electrodynamics

Three families of exact solutions for 2-dimensional gravity minimally coupled to electrodynamics are obtained in the context of $\mathcal{R}=T$ theory. It is shown, by supersymmetric formalism of quantum mechanics, that the quantum dynamics of a neutral bosonic particle on static backgrounds with both varying curvature and electric field is exactly solvable.