  We prove that any spectral sequence obeying a certain growth law is the
quantum spectrum of an equivalence class of classically integrable non-linear
oscillators. This implies that exceptions to the Berry-Tabor rule for the
distribution of quantum energy gaps of classically integrable systems, are far
more numerous than previously believed. In particular we show that for each
finite dimension $k$, there are an infinite number of classically integrable
$k$-dimensional non-linear oscillators whose quantum spectrum reproduces the
imaginary part of zeros on the critical line of the Riemann zeta function.
