  Real-analytic Levi-flat codimension two CR singular submanifolds are a
natural generalization to ${\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in
${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of
mixed-holomorphic equations with one variable antiholomorphic. We classify the
codimension two Levi-flat CR singular quadrics, and we notice that new types of
submanifolds arise in dimension 3 or greater. In fact, the nondegenerate
submanifolds, i.e. higher order purturbations of
$z_m=\bar{z}_1z_2+\bar{z}_1^2$, have no analogue in dimension 2. We prove that
the Levi-foliation extends through the singularity in the real-analytic
nondegenerate case. Furthermore, we prove that the quadric is a (convergent)
normal form for a natural large class of such submanifolds, and we compute its
automorphism group. In general, we find a formal normal form in
${\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal
invariants.
