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.