Solutions in Liouville theory on dS and AdS backgrounds

We find exact solutions of a Liouville-type scalar field theory on $D$-dimensional de Sitter and anti-de Sitter backgrounds, treating the geometry as nondynamical. Using the embedding-space formalism with an auxiliary null vector, we derive first-order (Bäcklund-like) equations whose integrability yields the solution of the Liouville equation for the scalar field. This method produces two classes of analytic solutions, whose physical properties are different for de Sitter and anti-de Sitter spacetimes.