A Hölder continuous vector field tangent to many foliations

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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 C¹ 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 to. We also show the existence of a continuous vector field X on ² and two foliations F and G on ² each tangent to X with a dense subset E of ² such that at every point x∈ E the leaves Fₓ and Gₓ of the foliation F and G through x are topologically transverse.