Partial conservation law in a schematic single j shell model

We report the discovery of a partial conservation law obeyed by a schematic Hamiltonian of two protons and two neutrons in a j shell. In our Hamiltonian, the interaction matrix element of two nucleons with combined angular momentum J is linear in J for even J and constant for odd J. It turns out that in some stationary states, the sum of the angular momenta J(p) and J(n) of the proton and neutron pairs is conserved. The energies of these states are given by a linear function of J(p) + J(n). The systematics of their occurrence is described and explained.