The Elliott scheme and its generalization

A study is made of the symmetry propertiea of the quadrupole interaction in the field of a spherically symmetric oscillator. The noninvarlance group of the Hamiltonian describing this interaction is found to be $SO(3,2)$ for a two-dimensional and $Sp(3,3)$ for a three-dlmonslonal operator. It is shown that if vibrations of the generalized model are present at all in the standard model with pairing and quadrupole forces they only appear as a result of competition between these forces.