  We construct an example of a H\"older continuous vector field on the plane
which is tangent to all foliations in a continuous family of pairwise distinct
$C^1$ foliations. Given any $1 \le r <\infty,$ the construction can be done in
such a way that each leaf of each foliation is the graph of a $C^r$ function
from $\R$ to $\R.$ We also show the existence of a continuous vector field $X$
on $\R^2$ and two foliations $\cal{F}$ and $\cal{G}$ on $\R^2$ each tangent to
$X$ with a dense subset $\cal E$ of $\R^2$ such that at every point $x\in \cal
E$ the leaves $F_x$ and $G_x$ of the foliation $\cal{F}$ and $\cal{G}$ through
$x$ are topologically transverse.
