Quantum spectra of classically chaotic systems without time reversal invariance

A hamiltonian with two degrees of freedom displays a transition from ordered to strongly chaotic classical motion under either of two perturbations, one symmetric, the other antisymmetric under time reversal. We compute numerically the eigenvalues of the quantum hamiltonians and find that each perturbation alone leads to the level statistics of the gaussian orthogonal ensemble. Only if both perturbations individually produce strongly chaotic classical behaviour, do we find the level statistics of the gaussian unitary ensemble.