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Theory of the quantal level statistics of classically integrable system,developed by Makino et al. in order to investigate the non-Poissonian behaviorsof 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 statisticalobservables of quantal levels in addition to LSD and LNV. In the theory ofMakino et al., the eigenenergy level is regarded as a superposition ofinfinitely many components whose formation is supported by the Berry-Robnikapproach in the far semiclassical limit\cite{Robn1998}. We derive the limiting$E(K,L)$ function in the limit of infinitely many components and elucidates itsproperties when energy levels show deviations from the Poisson statistics.

E(k,L) level statistics of classically integrable quantum systems based on the Berry-Robnik approach