We report an unexpected systematic degeneracy between different multiplets
in an inversion symmetric system of two coupled Gaudin models with
homogeneous couplings, as occurring for example in the context of solid state
quantum information processing. We construct the full degenerate subspace
(being of macroscopic dimension), which turns out to lie in the kernel of the
commutator between the two Gaudinmodels and the coupling term. Finally we
investigate to what extent the degeneracy is related to the inversion symmetry
of the system and find that indeed there is a large class of systems showing the
same type of degeneracy.