  We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin
model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two
interpretations of such solutions: as critical points of a cyclotomic master
function, and as critical points with cyclotomic symmetry of a certain
"extended" master function. In finite types, this yields a correspondence
between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model
and those of an "extended" non-cyclotomic Gaudin model. We proceed to define
populations of solutions to the cyclotomic Bethe equations, in the sense of
[Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163,
math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the
case of type A with the diagram automorphism, we associate to each population a
vector space of quasi-polynomials with specified ramification conditions. This
vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate
bilinear form which is (skew-)symmetric on the even (resp. odd) graded
subspace. We show that the population of cyclotomic critical points is
isomorphic to the variety of isotropic full flags in this space.
