Complete integrability, gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations

The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the $M$-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the $L$-operators.