  In this paper we consider germs of smooth Levi flat hypersurfaces, under the
following notion of local equivalence: S_1 ~ S_2 if their one-sided
neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a
simple invariant for this relation, which allows to prove some
characterizations of triviality (i.e. equivalence to the hyperplane). Then, we
employ the same invariant to construct infinitely many non-trivial classes,
including an infinite family of not equivalent hypersurfaces which are almost
everywhere analytic.
