One rank distribution over a smooth manifold Let $M$ be a smooth manifold and $D$ a rank one distribution there defined,i.e a rank one subbundle of the tangent bundle $TM$. 
Locally,this is nothing but a non vanishing vector field. But globally? Does it always exist a vector field whose span is the distribution or someone can give me an example where there isn't this global defined field?
 A: Here is a blueprint to generate examples of distributions which are not spanned by a global vectorfield: Assume $\dim M\ge 2$, take a loop $c$ in $M$ and define a vector field $X$ along $c$ which performs a rotation by $\pi$ as you move along $c$. That way $X$ will only be continuous at $c\backslash \{p_0\}$ for some point $p_0$, where it flips to the opposite direction. Then $X$ still defines a smooth distribution $D_0$ along $c$ and the question is whether this can be extended to a distribution $D$ on all of $M$. If this is the case, then $D$  cannot be spanned by a globally continuous vector field.
In general there is topological obstruction to extending $D_0$ (see below) - possibly  the only obstruction, but that I do not know. However it's easy now to cook up a particular example of what you are looking for:
Take $M$ the two-torus and $c$ a non-trivial loop, then $D_0$ can simply be extended by making making it constant along the perpendicular loops. (This shouldn't be hard to make rigorous). 

The topological obstruction:
If $M$ is simply-connected (or more generally if $\pi_1(M)$ does not have any index 2 subgroups), then $D$ comes from a global vector field. To see this, view $D\rightarrow M$ as a vector-bundle itself and consider the 'sphere-bundle' $SD\rightarrow M$ (say you put a metric on $M$, then $SD_x:=\{(x,\xi)\in T_xM\vert ~\xi\in D_x, \vert \xi  \vert =1\})$. Since $D$ has rank $1$, the fibre $SD_x$ consists of exactly two points, hence $SD\rightarrow M$ is a covering space. Simply connected spaces do not have any connected coverings, hence $SD\cong M \coprod M$ and in particular there exists a global section of $SD\rightarrow M$, in other words a nowwhere vanishing vector field that integrates $D$.
A: By the "hairy ball theorem" such a distribution can for example not be globally defined on any  $2n$-sphere.
