What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property:



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> Not only $D$ is not integrable but also there is no a two-dimensional foliation $F$ of $\mathbb{R}^4$ such that for every $x\in \mathbb{R}^4$, the tangent space to each leaf at $x$ is contained in $D\_x$.
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A strongly non-integrable distribution
Now that [this other question](https://mathoverflow.net/questions/294517/is-there-a-2-dimensional-foliation-tangent-to-this-particular-3-dimensional "this question") as been answered, we can answer this original question: The answer is 'yes' there exists a non-integrable tangent $3$-plane field on $\mathbb{R}^4$ that has no subordinate co-dimension $2$-foliation. 


It suffices to take an open $p$-neighborhood $U$ diffeomorphic to $\mathbb{R}^4$ of a point $p$ that lies on the line $L$ described in my answer to that question. Then choosing a diffeomorphism $\phi:\mathbb{R}^4\to U$, set $\beta = \phi^\*\alpha$. Then $\ker\beta$ will be a non-integrable tangent $3$-plane field on $\mathbb{R}^4$ that has no subordinate co-dimension $2$-foliation, because there is no such foliation on a neighborhood of $\phi^{-1}(p)$.


Of course, this says nothing about the following more precise question: Let $\beta$ be a $1$-form on $\mathbb{R}^4$ such that $\beta\wedge\mathrm{d}\beta$ is nowhere vanishing. Must there exist a co-dimension $2$ foliation of $\mathbb{R}^4$ whose leaves are all $\beta$-null?


