Let $M$ be a smooth manifold.

A *smooth distribution* $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector fields on $M$ satisfying $D_p = \text{span}\{X_p : X\in \mathcal C \} $ for every $p \in M$.

Remark that we do not ask the dimension of the fiber $D_p $ to be constant: we call a distribution *regular* if the dimension is constant, and *singular* if it is not.

We call a distribution $D$ *integrable* if for every point $p \in M$ there is a submanifold $S\subseteq M$ which is tangent to $D$ and satisfies $T_q S = D_q $ for every $q \in S$. In this case, it can be proved that the maximal connected integral manifolds of the distribution form a partition of $M$ into weakly embedded submanifolds of $M$, which we call the *foliation* associated to $D$.

Typical examples of integrable distributions are given by Lie algebroids: if $A\to M$ is a Lie algebroid over $M$ with anchor map $\rho : A\to TM$, then the image of $\rho $ is an integrable distribution on $M$. For example, every Poisson manifold has an integrable, possibly singular distribution given by the image of the Poisson bivector field $\Pi : T^*M\to TM $, and the induced foliation is precisely the symplectic foliation of the Poisson manifold.

The integrability problem for regular distributions is solved by the Frobenious theorem: a regular distribution $D$ is integrable if and only if it's involutive. A singular version of the Frobenious theorem can be stated in the following way: *a (possibly singular) distribution $D$ is integrable if and only if there is a family of vector fields $\mathcal C$ which span $D$ pointwise, such that the flow of every element of $\mathcal C$ preserves $D$* (see Theorem 3.5.10 of this book for a more precise statement and a proof).

A sufficient condition for the integrability of a singular distribution $D$ is the following: *there exists a module $\mathcal C$ of compactly supported vector fields spanning $D$ which is locally finitely generated and involutive*. Some people calls such an object a *Stefan-Sussman foliation*.

I have two related questions:

1) Is it true that every integrable distribution is spanned by a module $\mathcal C$ of compactly supported vector fields which is locally finitely generated and involutive?

2) Is it true that every integrable distribution is the image of the anchor map of some Lie algebroid?

Clearly, (2) implies (1). There is people which believe that (2) is true, and I would like to know if this question is still open.

Thank you!

for all$q\in S$ we have $T_q S = D_q $. With this definition,you can immediately see that my counterexample is not integrable. However, a set of generators of $D$ is $\partial / {\partial x},f \partial/{\partial y} $ where $f:\mathbb R ^2 \to \mathbb R $ is a smooth function which is zero for $x\leq 0$ and nonzero for $x>0$. $\endgroup$ – Ervin Mar 27 '17 at 17:29