Completeness of Bethe's states for generalized $XXZ$ model
  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 $A$. 
Anatol N. Kirillov and Nadejda A. Liskova