On a unified formulation of completely integrable systems

The purpose of this article is to show that a ${\mathcal{C}}^1$ differential system on ${\mathbb{R}}^n$ which admits a set of $n-1$ independent ${\mathcal{C}}^2$ conservation laws defined on an open subset $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, is essentially ${\mathcal{C}}^1$ equivalent on an open and dense subset of $\mathrm{\Omega }$, with the linear differential system $u^\prime_1=u_1, \ u^\prime_2=u_2,…,
u^\prime_n=u_n$. 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.