  We resolve a long-standing riddle in quantum chaos, posed by certain fully
chaotic billiards with constant negative curvature whose periodic orbits are
highly degenerate in length. Depending on the boundary conditions for the
quantum wave functions, the energy spectra either have uncorrelated levels
usually associated with classical integrability or conform to the "universal"
Wigner-Dyson type although the classical dynamics in both cases is the same.
The resolution turns out surprisingly simple. The Maslov indices of orbits
within multiplets of degenerate length either yield equal phases for the
respective Feynman amplitudes (and thus Poissonian level statistics) or give
rise to amplitudes with uncorrelated phases (leading to Wigner-Dyson level
correlations). The recent semiclassical explanation of spectral universality in
quantum chaos is thus extended to the latter case of "pseudo-arithmetical"
chaos.
