Bethe ansatz for the Gaudin model and its relation with Knizhnik–Zamolodchikov equations

We recall the construction of the common eigenvectors of Gaudin Hamiltonians based on the Bethe ansatz. In the case of an arbitrary Lie algebra, this construction can be done either recursively or explicitly and we prove the equivalence of the two methods. We also prove that Bethe vectors are singular only if the Bethe equations are satisfied. In each eigenspace of the spin operator we construct additional common eigenvectors, having the same eigenvalue as the vacuum vector and which can not be obtained by the Bethe ansatz. These eigenvectors are not singular. We also recall the connection between Bethe vectors and integral solutions of the KZ equation. In an analogous way, the additional vectors lead to solutions of KZ equation which are not singular vectors and do not have an integral representation.