  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 i s, 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 8 sites, and find some support for the
conjecture. We also exhibit several examples of strange solutions.
