Consider the following example of a distribution (given here):
I tried to draw this. If $p=(a,b,c)$ then $$ X_p = (1,0,-b), Y_p = (0,1,0)$$
Then the planes in the distribution are planes spanned by $X_p,Y_p$.
We see that the plane spanned by $X_p,Y_p$ is a plane that rotates around the vector $Y_p$ as $p$ moves along the $y$-axis.
Assume we had a surface $S$ that was tangent to all this twirling planes. Without loss of generality, assume the surface is located in $\mathbb R^3$ such that the origin is on the surface.
Then we have a plane, coincidentally parallel to the $xy$-plane, that is tangent to $S$ at $0$. In other words: the $xy$-plane is tangent to $S$.
This is as far as I can follow the explanation given in the text. But everything that follows I do not understand.
For example, only because the $xy$-plane is tangent to $S$ at $0$ it is not clear to me why $S$ would intersect the $x$-axis in a line segment (for example, $S^2$ can be tangent to the $xy$-plane an does not intersect the $x$-axis in a line segment).
But even if this was clear to me and I assume that $S$ intersects this axis in a line segment the rest of the explanation is also not clear to me: travelling along an intersection axis does not seem to contradict that the planes are twisting.
Please could someone explain this to me?