The Completely Integrable Differential Systems are Essentially Linear Differential Systems

© 2015, Springer Science+Business Media New York. Let x˙=f(x) be a C^k autonomous differential system with k∈N∪{∞,ω} defined in an open subset Ω of Rⁿ. Assume that the system x˙=f(x) is C^r completely integrable, i.e., there exist n-1 functionally independent first integrals of class C^r with 2≤r≤k. As we shall see, we can assume without loss of generality that the divergence of the system x˙=f(x) is not zero in a full Lebesgue measure subset of Ω. Then, any Jacobian multiplier is functionally independent of the n-1 first integrals. Moreover, the system x˙=f(x) is C^r-1 orbitally equivalent to the linear differential system y˙=y in a full Lebesgue measure subset of Ω. Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.