  The notion of S-stability of foliations on branched simple polyhedrons is
introduced by R. Benedetti and C. Petronio in the study of characteristic
foliations of contact structures on 3-manifolds. We additionally assume that
the 1-form $\beta$ defining a foliation on a branched simple polyhedron $P$
satisfies $d\beta>0$, which means that the foliation is a characteristic
foliation of a contact form whose Reeb flow is transverse to $P$. In this
paper, we show that if there exists a 1-form $\beta$ on $P$ with $d\beta>0$
then we can find a 1-form with the same property and additionally being
S-stable. We then prove that the number of simple tangency points of an
S-stable foliation on a positive or negative flow-spine is at least 2 and give
a recipe for constructing a characteristic foliation of a 1-form $\beta$ with
$d\beta>0$ on the abalone.
