  We study previously un-researched second order statistics - correlation
function of spectral staircase and global level number variance - in generic
integrable systems with no extra degeneracies. We show that the global level
number variance oscillates persistently around the saturation spectral
rigidity. Unlike other second order statistics - including correlation function
of spectral staircase - which are calculated over energy scales much smaller
than the running spectral energy, these oscillations cannot be explained within
the diagonal approximation framework of the periodic orbit theory. We give
detailed numerical illustration of our results using four integrable systems:
rectangular billiard, modified Kepler problem, circular billiard and elliptic
billiard.
