  We present new $n=(1,1)$ 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 B\"acklund transformations associated with the supersymmetric non--linear
equations.
