We present an interesting reformulation of a collection of dilaton gravity models in two spacetime dimensions into a field theory of two decoupled Liouville fields in flat space, in the presence of a Maxwell gauge field. An effective action is also obtained, encoding the dynamics of the dilaton field and the single gravitational degree of freedom in a decoupled regime. This effective action represents an interesting starting point for future work, including the canonical quantization of these classes of nontrivial models of gravity-coupled matter systems.
Zonetti, S., & Govaerts, J. (2012). Duality between (1+1)-dimensional dilaton–Maxwell gravity and Liouville field theory. Journal of Physics A: Mathematical and Theoretical, 45(4), 42001. https://doi.org/10.1088/1751-8113/45/4/042001 (Original work published 2012)