On the CR-curvature of Levi degenerate tube hypersurfaces

In our recent article (to appear in the Journal of Differential Geometry in 2016) we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we discovered that for the CR-curvature of such a hypersurface to vanish it suffices to require that only two coefficients (called $\Theta^2_{21}$ and $\Theta^2_{10}$) in the expansion of a single component of the CR-curvature form be identically zero. In this paper, we show that, surprisingly, the vanishing of the entire CR-curvature is in fact implied by the vanishing of a single quantity derived from $\Theta^2_{10}$. This fact strengthens the main theorem of the earlier article and also leads to a remarkable system of partial differential equations. Furthermore, we explicitly characterize the class of not necessarily CR-flat tube hypersurfaces given by the vanishing of $\Theta^2_{21}$.