New supersymmetric generalization of the Liouville equation

We present n = (1,1) and n = (1,0) supersymmetric generalization of the Liouville equation which originate from a geometrical approach to describing the classical dynamics of Green-Schwarz superstrings in N = 2, D = 3 and N = 1, D = 3 target superspace. Considered are a zero curvature representation and Bäcklund transformations associated with the supersymmetric non-linear equations.