  We study the Bethe ansatz equations for a generalized $XXZ$ model on a
one-dimensional lattice. Assuming the string conjecture we propose an integer
version for vacancy numbers and prove a combinatorial completeness of Bethe's
states for a generalized $XXZ$ model. We find an exact form for inverse matrix
related with vacancy numbers and compute its determinant. This inverse matrix
has a tridiagonal form, generalizing the Cartan matrix of type $A$.
