A Hölder continuous vector field tangent to many foliations

We construct an example of a Hölder continuous vector field on the plane which is tangent to all foliations in a continuous family of pairwise distinct C foliations. Given any 1 ≤ r < ∞, the construction can be done in such a way that each leaf of each foliation is the graph of a Cr function from R to R. We also show the existence of a continuous vector field X on R and two foliations F and G on R each tangent to X with a dense subset E of R such that at every point x ∈ E the leaves Fx and Gx of the foliation F and G through x are topologically transverse.