Taken individually, the Slater determinants Ψ N I [see Eq. ( )] preserve the third projections S z and V z , but not necessarily the square Ŝ2 and V2 of the total spin and total isospin. However, because Ŝ2 and V2 commute with the many-body Hamiltonians H MB and H MB + H VS , the associated exact many-body solutions are eigenstates of both Ŝ2 and V2 with eigenvalues S(S + 1) and V(V + 1),