Let $M$ be a smooth manifold and $G\_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G\_k(M)$ be a continuous distribution on $K$.


We say $E$ is integrable on $K$ if there exists a foliation $\mathcal{F}$ (or lamination, since it may only foliates a subset of $M$), such that $T\_x\mathcal{F}(x)=E\_x$. 


My quesion is: if $(E,K)$ is integrable, will there exist an open neighborhood $U\supset K$ that admits an integrable extension $\widetilde{E}:U\to G\_k(M)$?




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For example $K$ is a hyperbolic invariant set of a diffeomorphism $f:M\to M$. It is known that there are stable and unstable foliations (manifolds) through $K$. I donot know if we can extend the foliations to an open neighborhood of $K$.



Extension of integrable distribution over a subset
The answer in general is no. 


If $K$ is a submanifold of $M$ then tangent bundle of $K$ defines an integrable distribution on $K$. To wit we are talking about the foliation with just one leaf: $K$.


If we can extend this foliation to a neighborhood of $K$ then the restriction of Bott's connection to $K$ induces a flat connection on the normal bundle of $K$. This imposes restrictions on $K$. For instance $K$ cannot be a projective curve in $\mathbb P^2(\mathbb C)$.




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Recall that Bott's connection is the partial connection on the normal bundle of $\mathcal F $ defined as follows. If $T\mathcal F$ is the tangent bundle of a foliation and $N\mathcal F$ is its normal bundle then they fit into the exact sequence:
$$
0 \to T \mathcal F \to TM \stackrel{\pi}{\to} N\mathcal F \to 0 .
$$ 
Now, if $v$ is a local section of $T\mathcal F$ and $w$ is a local section of $N\mathcal F$
then Bott's partial connection $\nabla : T\mathcal F \times N\mathcal F \to N \mathcal F$ is defined by the formula  

$$
\nabla\_{v}(w) = \pi ([v, \pi^{-1}(w)])
$$
where $\pi^{-1}(w)$ is an arbitrary lifting of $w$ to $TM$. The involutiviness of $T\mathcal F$ implies that $\nabla$ is well-defined. It is not hard to check that $\nabla^2=0$. 


