  The note deals with the Gaudin model associated with the tensor product of n
irreducible finite-dimensional sl_{N+1}-modules marked by distinct complex
numbers z_1,..., z_n. The Bethe Ansatz is a method to construct common
eigenvectors of the Gaudin hamiltonians by means of chosen singular vectors in
the factors and z_j's. These vectors are called Bethe vectors.
  The question if the Bethe vectors are non-zero vectors is open. By the
moment, the only way to verify that was based on a relation to critical points
of the master function of the Gaudin model, and non-triviality of a Bethe
vector was proved only in the case when the corresponding critical point is
non-degenerate ([ScV], [MV1]). However degenerate critical points do appear in
the Gaudin model (see Section12 of [ReV]).
  We believe that the Bethe vectors never vanish, and suggest an approach that
does not depend on non-degeneracy of the corresponding critical point. The idea
is for a Bethe vector to choose a suitable subspace in the weight space and to
check that the projection of the Bethe vector to this subspace is non-zero. We
apply this approach to verify non-triviality of Bethe vectors in new examples.
