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)
