  A second order finite-difference equation has two linearly independent
solutions. It is shown here that, like in the continuous case, at most one of
the two can be a polynomial solution. The uniqueness in the classical
continuous Heine-Stieltjes theory is shown to hold under broader hypotheses
than usually presented. A difference between regularity condition and
uniqueness is emphasized. Consistency of our uniqueness results is also checked
against one of the Shapiro problems. An intrinsic relation between the
Heine-Stieltjes problem and the discrete Bethe Ansatz equations allows one to
immediately extend the uniqueness result from the former to the latter. The
results have implications for nondegeneracy of polynomial solutions of physical
models.
