  We consider non-vanishing boundary conditions (NVBC) for the NLS model
[6,7,27] in the context of the hybrid dressing transformation and
$\tau$-function approach. In order to write the NLS model in a suitable form to
deal with non-vanishing boundary conditions it is introduced a new spectral
parameter in such a way that the usual NLS parameter will depend on the affine
parameter through the so-called Zukowsky function. In the context of the
dressing transformation the introduction of the affine parameter avoids the
construction of certain Riemann sheets for the usual NLS spectral parameter. In
this way one introduces a Lax pair defined for the new spectral parameter and
the relevant NVBC NLS $\tau$ functions are obtained by the dressing
transformation method. We construct the one and two dark-soliton solutions
explicitly.
