Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by
$$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$
In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have a first integral. But this is for discrete DS.
My question is the following: Is there a similar result for continuous time DS as defined above?
Or more formally, is it known that it is not $\mathcal{C}^1$ generic to have an invariant subset $U \subset M$ (non-trivial, i.e. $U$ itself has an open subset) such that on $U$ there exists a first integral.
(It is clear to me that by the straightening theorem for vector fields we alawys have an open set with a first integral but this set is not invariant.)
