Degenerate eigenvalues for Hamiltonians with no obvious symmetries

Certain Hamiltonians based on two coupled quantum mechanical spins exhibit degenerate eigenvalues despite having no obvious symmetries: the corresponding conserved quantities do not appear to have a simple form when expressed as polynomials of the generators of rotations for the respective spins. As observed in [1], one such Hamiltonian helps explain resonances in the spin relaxation rate of optically pumped Rb_2, as a function of applied magnetic field. For this Hamiltonian and others closely related to it, we give an explanation why the degeneracies exist, based on an argument inspired by supersymmetry