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.
We derive a canonical form for smooth vector fields on ℜn+1. We use this to demonstrate the local multi-Hamiltonian nature of the corresponding flows. Associated with the canonical form is an inhomogeneous linear PDE whose solutions provide conserved measures. These can be used to construct the local Hamiltonians.
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