  We prove that a singular real-analytic Levi-flat hypersurface $H$ in $\mathbb
C^n$ being Segre-degenerate at a point $p$ is equivalent to the existence of a
so-called support curve, that is, a holomorphic curve that intersects $H$ at
exactly one point, which in turn is equivalent to the existence of support
curves on at least two sides of $H$ at $p$. The existence of such two-sided
support provides families of analytic discs attached to $H$ that covers a
neighborhood of $p$. The existence of such discs has two corollaries. First,
any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to
a fixed neighborhood of $p$. Second, the rational hull of $H$ is a neighborhood
of $p$, and thus no Levi-flat Segre-degenerate hypersurface in $\mathbb C^n$
can be rationally convex.
