  Theory of the quantal level statistics of classically integrable system,
developed by Makino et al. in order to investigate the non-Poissonian behaviors
of level-spacing distribution (LSD) and level-number variance
(LNV)\cite{MT03,MMT09}, is successfully extended to the study of $E(K,L)$
function which constitutes a fundamental measure to determine most statistical
observables of quantal levels in addition to LSD and LNV. In the theory of
Makino et al., the eigenenergy level is regarded as a superposition of
infinitely many components whose formation is supported by the Berry-Robnik
approach in the far semiclassical limit\cite{Robn1998}. We derive the limiting
$E(K,L)$ function in the limit of infinitely many components and elucidates its
properties when energy levels show deviations from the Poisson statistics.
