A normal form of completely integrable systems

The purpose of this article is to show that a C1C1 differential system on RnRn which admits a set of n−1n−1 independent C2C2 conservation laws defined on an open subset Ω⊆RnΩ⊆Rn, is essentially C1C1 equivalent on an open and dense subset of ΩΩ, with the linear differential system u1′=u1,u2′=u2,…,un′=un. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka–Volterra system, and respectively the Euler equations from the free rigid body dynamics.