  To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak
g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large
commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy
of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the
special case $\sigma = \text{id}$.
  We go on to construct joint eigenvectors and their eigenvalues for this
hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain
consisting of a tensor product of Verma modules. To do so we generalize an
approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving
vertex algebras and the Wakimoto construction. As part of this construction, we
make use of a theorem concerning cyclotomic coinvariants, which we prove in a
companion paper.
  As a byproduct, we obtain a cyclotomic generalization of the
Schechtman-Varchenko formula for the weight function.
