The Lie algebra of vector fields and the action of the associated Lie group

Question

Let $M$ be a smooth manifold and let $\mathfrak g$ be the Lie algebra of smooth vector fields on $M$. Suppose that $\mathfrak g$ is finite dimensional.

Let $G$ be a connected Lie group such that $\mathfrak g$ be its Lie algebra.

Are the following statements correct:

1. There is an induced group action of $G$ on $X$ if the vector fields in $\mathfrak g$ are integrable.
2. If $G$ acts on $X$ transitively then the vector fields in $\mathfrak g$ are integrable.