Let S ⊂ ℂⁿ, n ≥ 3, be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: a sphere with two horns, elementary models and their gluings. Some particular cases of S being a graph are also described.

Every homogeneous circular convex domain $D\subset {\mathbb{C}}^n$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G=Aut\left(D\right)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K\subset GL\left(n,\mathbb{C}\right)$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of ${\mathbb{C}}^n$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested...