  Along the line of thoughts of Berry and Robnik\cite{[1]}, we investigated the
gap distribution function of systems with infinitely many independent
components, and discussed the level-spacing distribution of classically
integrable quantum systems. The level spacing distribution is classified into
three cases: Case 1: Poissonian if $\bar{\mu}(+\infty)=0$, Case 2: Poissonian
for large $S$, but possibly not for small $S$ if $0<\bar{\mu}(+\infty)< 1$, and
Case 3: sub-Poissonian if $\bar{\mu}(+\infty)=1$. Thus, even when the energy
levels of individual components are statistically independent, non-Poisson
level spacing distributions are possible.
