The completely integrable differential systems are essentially linear differential systems (Q496306)

This paper is concerned with the autonomous differential system \[ \dot x=f(x) \] on an open subset \(\Omega\subseteq\mathbb{R}^k\) with \(f\in C^k\), \(k\in\mathbb{N}\cup\{\infty,\omega\}\). Assuming that the system is \(C^r\)-completely integrable, \(2\leq r\leq k\), (that is there exist \(n-1\) functionally independent \(C^r\) first integrals), the authors prove that {\parindent=6mm \begin{itemize}\item[(a)] The system has a \(C^{r-1}\) Jacobian multiplier (that is a function \(J\) such that \(\mathrm{div}(Jf)\equiv 0\)) defined in a subset of \(\Omega\) of full Lebesgue measure; \item[(b)] If \(J\) is a \(C^{r-1}\) Jacobian multiplier of the system, then \(J\) is functionally independent of the first integrals; \item[(c)] There exist a subset \(\Omega_0\subseteq\Omega\) of full Lebesgue measure in which the system is \(C^{r-1}\) orbitally equivalent to the system \(\dot y = y\). \end{itemize}} Statement (a) had been, in fact, proved by one of the authors in a previous paper but the proof is provided for completeness. The paper closes with two applications to important equations, namely the system of Rössler and that of May-Leonard (a 3-dimensional generalization of Lotka-Volterra.