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Regular version of the site
Gaudin algebra is the commutative subalgebra in $U(\g)^{\otimes n}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\g$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_n$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\g$-modules, for all values of the parameters. The corollary of this result is the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\g$-modules are in 1-1 correspondence with monodromy-free opers on the projective line with regular singularities at the points $z_1,\ldots,z_n,\infty$ and prescribed residues at singular points.