States with v1=lambda, v2=-lambda and reciprocal equations in the   six-vertex model
  The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,...,vn, the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary. The number obtained is independent of the value of Delta, but depends on N. We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a,b,c such eigenvalues are real. 
M. J. Rodriguez-Plaza