  We prove that a transformation, conjectured in our previous work, between
phase-space variables in $\s$-models related by Poisson-Lie T-duality is indeed
a canonical one. We do so by explicitly demonstrating the invariance of the
classical Poisson brackets. This is the first example of a class of $\s$-models
with no isometries related by canonical transformations. In addition we discuss
generating functionals of canonical transformations in generally non-isometric,
bosonic and supersymmetric $\s$-models and derive the complete set of
conditions that determine them. We apply this general formalism to find the
generating functional for Poisson-Lie T-duality. We also comment on the
relevance of this work to D-brane physics and to quantum aspects of T-duality.
