Singular solutions, repeated roots and completeness for higher-spin chains

We investigate the completeness of the solutions of the Bethe equations for the integrable spin-s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string is, i(s - 1), ..., -i(s - 1), -is are singular. For s > 1/2, there exist also 'strange' solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s = 1 and s = 3/2 up to eight sites, and find some support for the conjecture. We also present several examples of strange solutions.