I try to solve this exercise:
The integral curves of a vector field are immersed submanifolds of the manifold M, but not necessarily submanifolds whick form a decomposition of M.
Lets talk about the second part.
Do the maximal integral curves form always a decomposition of M? I think yes, because they have to be disjoint.
How can it be, that integral curves doesn't form a decomposition of the Manifold M?