Widening the Scope of a Partial Dynamical Symmetry

In a single j shell calculation in which only T=1(even J) two-particle matrix elements were non-zero there was a partial dynamical symmetry e.g. for 2 protons and 2 neutrons in the f$_{7/2}$ shell there is a degeneracy of states with angular momenta I=3,7,9,10. These have non -zero components only for (J$_{p}$, J$_{n}$) =(4,6) or (6,4). These I's cannot occur for 4 identical partices ($^{44}$ Ca).We then consider a ``123'' interaction which for J=0 to 7 is (0,0,1,0,2,0,3,0). Then I=6 and I=8 also come into play. Fot these (J$_{p}$+J$_{n}$ ) is a good quantum number .One gets an equally spaced multidegerate levels ( ``vibrational spectra'') with separation of 1.5 MeV. Each of these levels has fixed (J$_{p}$ +J$_{n}$ . For (J$_{p}$+J$_{n}$) equal to 6 we have I=3 an I= 6 as ;for 8 we get 6,7 ,8; for 10,we get 3,7,9,10 and for 12 we get 10,12. In the g$_{9/2}$ shell with a ``1234''(J$_{p}$ +J$_{n}$) ranges from 8 to 16 and in h$_{11/2}$with ``12345'' from 10 and 20.