The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Abstract

Let $$\dot{x}=f(x)$$ be a $$C^k$$ autonomous differential system with $$k\in {\mathbb {N}}\cup \{\infty ,\omega \}$$ defined in an open subset $$\varOmega $$ of $${\mathbb {R}}^n$$ . Assume that the system $$\dot{x}=f(x)$$ is $$C^r$$ completely integrable, i.e., there exist $$n-1$$ functionally independent first integrals of class $$C^r$$ with $$2\le r\le k$$ . As we shall see, we can assume without loss of generality that the divergence of the system $$\dot{x}=f(x)$$ is not zero in a full Lebesgue measure subset of $$\varOmega $$ . Then, any Jacobian multiplier is functionally independent of the $$n-1$$ first integrals. Moreover, the system $$\dot{x}=f(x)$$ is $$C^{r-1}$$ orbitally equivalent to the linear differential system $$\dot{y}=y$$ in a full Lebesgue measure subset of $$\varOmega $$ . Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.