Bäcklund transformations for superextensions of the Liouville equation

A general technique is described for constructing Bäcklund transformations for the Liouville equation and its superextensions in the framework of a method based on nonlinear realizations of infinite-dimensional symmetries. Superfield Bäcklund transformations are found for the $N=2$ and $N=4$ supersymmetric Liouville equations together with the Bäcklund transformations to the solutions of the corresponding free equations. The geometrical meaning of these transformations is established.