  We establish the isomorphism between a nonlinear $\sigma$-model and the
abelian gauge theory on an arbitrary curved background, which allows us to
derive integrable models and the corresponding Lax representations from gauge
theoretical point of view. In our approach the spectral parameter is related to
the global degree of freedom associated with the conformal or Galileo
transformations of the spacetime. The B$\ddot{\rm a}$cklund transformations are
derived from Chern-Simons theory where the spectral parameter is defined in
terms of the extract compactified space dimension coordinate.
