  We continue our study, initiated in an earlier article, of a class of rigid
hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi
degenerate of rank 1, having zero CR-curvature. We drop the restrictive
assumptions of the earlier paper and give a complete description of the class.
Surprisingly, the answer is expressed in terms of solutions of several
well-known differential equations, in particular, the equation characterizing
conformal metrics with constant negative curvature and a nonlinear
$\bar\partial$-equation.
