The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Let be a autonomous differential system with defined in an open subset of . Assume that the system is completely integrable, i.e., there exist functionally independent first integrals of class with . As we shall see, we can assume without loss of generality that the divergence of the system is not zero in a full Lebesgue measure subset of . Then, any Jacobian multiplier is functionally independent of the first integrals. Moreover, the system is orbitally equivalent to the linear differential system in a full Lebesgue measure subset of . Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.