Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle 
$VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under 
the Lie bracket of $\Gamma(TM)$ or in other words if $[X,Y] \in \Gamma(VM)$ for all
$X,Y \in \Gamma(VM)$.


On the other side the Lie bracket of two vector field can be expressed entirely by the flow transformations of the fields, that is we have:


$$[X,Y] = \frac{1}{2}\frac{\partial^2}{\partial\_t^2} |\_{t=0}(Fl^Y \_{-t}\circ Fl^X \_{-t}\circ
Fl^Y \_{t}\circ Fl^X \_{t})$$


where $Fl^X$ and $Fl^Y$ are the flow transformations of $X$ and $Y$ respectively.


Now the question is, can (and if yes, how) we decide whether or not o subbundle of $TM$ is involutive entirely in terms of flow transformations?


If somehow we want to proof that the bracket is closed on a subbundle but we know very little about it but instead know much about the associated flow transformations. 



Criteria for Involutive Subbundles
The subbundle is involutive if and only if the image of any given $x \in M$ under all of the flow transformations is an immersed smooth submanifold of $M$ with dimension equal to the rank of the subbundle.


ADDED: One direction follows by the Frobenius theorem. The other direction is even easier.


