Gravitational dressing of massive soliton theories

The massive soliton theories describe integrable perturbations of WZW cosets as generalized multi-component sine-Gordon models. We study their coupling to 2-dim gravity in the conformal gauge and show that the resulting models can be interpreted as conformal non-Abelian Toda theories when a certain algebraic condition is satisfied. These models, however, do not provide quantum mechanically consistent string backgrounds in the case the underlying WZW constraints are first solved classically.