On the CR-curvature of Levi degenerate tube hypersurfaces

In article [I2] we studied tube hypersurfaces in C3 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 Θ2 21 and Θ2 10) in the expansion of a certain 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 Θ2 10. This result strengthens the main theorem of [I2] 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 Θ2 21.