2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187142We 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 $A$.20 pages, PlainTEXHigh Energy Physics - Theorytext