  We consider Reeb flows associated with tight contact forms on $S^3$. We study
the existence of a transverse foliation, called $3-2-1$ foliation, which is a
singular foliation of $S^3$ having exactly three binding orbits with
Conley-Zehnder indices $3$, $2$ and $1$. All the regular leaves of the
foliation are transverse to the Reeb flow. The existence of a homoclinic to the
binding orbits with Conley-Zehnder index $2$ follows from this foliation. Our
main results give sufficient conditions for the existence of a $3-2-1$
foliation with prescribed binding orbits. We also give an example of a
Hamiltonian flow on $\mathbb{R}^4$ admitting a $3-2-1$ foliation when
restricted to an energy level.
