Frobenius rank of a manifold

Question

The rank of an smooth manifold M is defined by Milnor, as follows:

"The maximum number of independent commuting vector fields on M"

For example it is well known that the rank of $S^{3}$ is 1 (Lima, On commuting vector fields on S^3, Annals of math,1965)

One can consider a generalization as follows:

The Frobenius rank of an n- manifold is:

The maximum number $k< n$ such that there are k independent smooth vector fields $X_{1},X_{2},\ldots,X_{k}$, globally defined on $M$, such that $[X_{i}, X_{j}](p) \in \text{Span}(X_{1}(p),\ldots X_{k}(p))$, for all $p\in M$ and for all $i,j\in \{1,2,\ldots,k\}$.

Now the question:

What is the Frobenius rank of $S^{3}$ or $S^{7}$?

In fact this question searchs for the maximum number $k$ such that $M$ admits a $k$ dimensional foliation which distribution can be generated by k global independent vector fields?

The motivation for this question is the question "Can the Reeb foliation be generated by two independent global vector fields?"

Answer 1

I think the Reeb foliation can be generated by two independent global vector fields so the Frobenius rank of $S^{3}$ is "2".

The reason is that every real vector bundle on $S^{3}$ is a trivial bundle. so the two dimensional subvector bundle of $TS^{3}$ tangent to the Reeb foliation, is a trivial bundle. So there are two global continuous sections for this distribution. (These two sections can be choosed smooth by standard approximations). This proves our claim.

The reason that every n-dim real bundle on $S^{3}$ is trivial:

Using Technics of clutching functions, explained in "K-theory and vector bundles" by Allen Hatcher we conclude that the equivalent classes of n dim vector bundles on $S^{k}$ is isomorphic the homotopy class of continuous functions from $S^{k-1}$ to $GL(n,\mathbb{R})$. In our particular case $S^{3},\;\;k=3,\;n=2$, we have that $\pi_{2}(GL(2,\mathbb{R}))$ is trivial, see Homotopy groups of Lie groups So the only n. dim real bundle on $S^{3}$ is trivial.

So why it is not customary to introduce the Reeb foliation by two explicit global vector fields tangent to $S^{3}$? On the other hand, it is well known that the Reeb foliation can not be generated by two Analytic vector fields. So it is natural to ask:

Is there a global analytic vector field tangent to the Reeb foliation?(We can consider both singular and non singular case. In the singular case, what would be the nature of singularities?)