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 $\widetilde{E}:U\to G_k(M)$?

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 $K$.