A family of solvable non-rational conformal field theories

We find non-rational conformal field theories in two dimensions, which are solvable due to their correlators being related to correlators of Liouville theory. Their symmetry algebra consists of the dimension-two stress-energy tensor, and two dimension-one fields. The theories come in a family with two parameters: the central charge c and a complex number m. The special case m = 0 corresponds to Liouville theory (plus two free bosons), and m = 1 corresponds to the H₃⁺ model. In the case m = 2 we show that the correlators obey third-order differential equations, which are associated to a subsingular vector of the symmetry algebra.