Let g be a classical Lie algebra. There is a family of commutative subalgebras in Ug over the Deligne-Mumford moduli space M0,n+1 with the following properties:
• over the subset of conﬁgurations of n points on A1 it coincides with the family Gaudin subalgebras,
• over the special points it coincides with Gelfand-Zeitlin subalgebras.
Moreover, over the real locus M0,n+1(R) it acts with simple spectrum in the multiplicity space of the tensor product of any n irreducible ﬁnite-dimensional g-modules. Thus, for any n + 1 dominant integer weights we have a covering of eigenlines over M0,n+1(R). The monodromy action of this covering gives us an action of the cactus group (=Sn-equivariant fundamental group of M0,n+1(R)) on the ﬁbers, which are parametrized by various kinds of Young tableaux over the special points. .