Frobenius rank of a manifold 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?"
 A: 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?)
