  We discover quantum Hall like jumps in the saturation spectral rigidity in
the semiclassical spectrum of a modified Kepler problem as a function of the
interval center. These jumps correspond to integer decreases of the radial
winding numbers in classical periodic motion. We also discover and explain
single harmonic dominated oscillations of the level number variance with the
width of the energy interval. The level number variance becomes effectively
zero for the interval widths defined by the frequency of the shortest periodic
orbit. This signifies that there are virtually no variations from sample to
sample in the number of levels on such intervals.
