We give necessary and sufficient conditions for the complete integrability of first order N–dimensional differential systems.

We propose a new method to determine in the Jacobi Theorem the last $N-1$ first integral for the complete integrability of an N–dimensional differential system with $N-2$ independent first integrals and with a Jacobi multiplier.

As an application we study the complete integrability of some 3–dimensional differential systems, more precisely the complete integrability of the asymmetric and symmetric May–Leonard differential systems.

中文翻译：

---

矢量场的完全可积性 ${\mathbb{电阻}}^N$

我们给出一阶N维微分系统完全可积的充要条件。

我们提出了一种新方法来确定雅可比定理中的最后一个 $N-1$N维微分系统的完全可积性的第一积分$N-2$ 独立的第一积分和雅可比乘数。

作为一个应用，我们研究了一些 3 维微分系统的完全可积性，更准确地说是非对称和对称 May-Leonard 微分系统的完全可积性。