was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone {x 2 1 +x 2 2 −x 2 3 = 0}. Another representation is the tube u = (Re z) 2 1−Re ζ . The Gaussier-Merker model has 7-dimensional rigid automorphisms group. Inspired by Alexander Isaev, we study rigid biholomorphisms: (z, ζ, w) −→ (f (z, ζ), g(z, ζ), ρw + h(z, ζ)) =: (z , ζ , w ).

The class IV 2 of 2-nondegenerate constant Levi rank 1 hypersurfaces M 5 ⊂ C 3 is governed by Pocchiola's two primary invariants W 0 and J 0 . Their vanishing characterizes equivalence of such a hypersurface M 5 to the tube M 5LC over the real light cone in R 3 . When either W 0 ≡ 0 or J 0 ≡ 0, by normalization of certain two group parameters c and e, an invariant coframe can be built on M 5 , showing that the dimension of the CR automorphism group drops from 10 to 5.This paper constructs an explicit {e}-structure in case W 0 and J 0 do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on a computer now appear in details, especially the determination of a secondary invariant R, expressed in terms of the first jet of W 0 . All other secondary invariants of the {e}-structure are also expressed explicitly in terms of W 0 and J 0 .

Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan’s method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$D real-analytic real hypersurface $M^{9}\subset \mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.