Completely Integrable Differential Systems Are Essentially Linear

Let ẋ = f(x) be a C autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R. Assume that the system ẋ = f(x) is C completely integrable, i.e. there exist n−1 functionally independent first integrals of class C with 2 ≤ r ≤ k. If the divergence of system ẋ = f(x) is non–identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the system ẋ = f(x) is Cr−1 orbitally equivalent to the linear differential system ẏ = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers.