  It was recently shown by Jimbo, Miwa and Smirnov that the correlation
functions of a generalized XXZ chain associated with an inhomogeneous
six-vertex model with disorder parameter $\alpha$ and with arbitrary
inhomogeneities on the horizontal lines factorize and can all be expressed in
terms of only two functions $\rho$ and $\omega$. Here we approach the
description of the same correlation functions and, in particular, of the
function $\omega$ from a different direction. We start from a novel multiple
integral representation for the density matrix of a finite chain segment of
length $m$ in the presence of a disorder field $\alpha$. We explicitly
factorize the integrals for $m=2$. Based on this we present an alternative
description of the function $\omega$ in terms of the solutions of certain
linear and nonlinear integral equations. We then prove directly that the two
definitions of $\omega$ describe the same function. The definition in the work
of Jimbo, Miwa and Smirnov was crucial for the proof of the factorization. The
definition given here together with the known description of $\rho$ in terms of
the solutions of nonlinear integral equations is useful for performing e.g.\
the Trotter limit in the finite temperature case, or for obtaining numerical
results for the correlation functions at short distances. We also address the
issue of the construction of an exponential form of the density matrix for
finite $\alpha$.
