# A non vanishing vector field on \$S^3\$ whose flow does not preserves any transversal foliation

Is there a non vanishing vector field $$X$$ on $$S^3$$ which does not admit a transversal $$2$$ dimensional foliation? if the answer is negative, is there a non vanishing vector field $$X$$ on $$S^3$$ which does not admit a transversal foliation whose leaves are invariant under the flow of $$X$$? If the answer is positive, is there an example of this situation with the extra assumption that $$X$$ is invariant under the obvious action of $$S^1$$ on $$S^3$$?

The motivation is described in the following post and the paper linked in that post.(The flow invariant foliation)

Irrational closed orbits of vector fields on \$ S^2\$ (Limit cycles and trace formula)