Consider the Hopf fiber of the $\mathbb{S}^3$ sphere over the $\mathbb{S}^2$ sphere. Consider the two-dimensional distribution in $\mathbb{S}^3$ that is orthogonal to the fibers of this fiber. Use Frobenius' theorem to show that this distribution is not completely integrable.
This problem is taking away my peace, I cannot develop anything plausible.
What I thought is that we have to determine a $1-$form $\omega$ in an open $\mathbb{S}^3$ (I think) that represents the differential system defined by the orthogonal distribution of the fibers, the idea would be to find a diffeomorphism $\varphi$ of a coordinate system in $\mathbb{S}^3$ defined by the canonical system. My interest is to show that $\mathrm{d} \omega \wedge \omega =0$, for this I use the fact that $\varphi$ is diffeomorphism, then it is equivalent to show that $\varphi^*(\mathrm{d} \omega \wedge \omega) =0$ (I think it gets easier). Then, doing some calculations we arrived at the conditions of integrability