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Normal Forms for Rigid $\mathfrak{C}_{2,1}$ Hypersurfaces $M^5 \subset \mathbb{C}^3$

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathcal{C}^ω$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, ζ, w = u + iv)$: \[ u = F\big(z,ζ,\bar{z},\barζ\big). \] The Gaussier-Merker model $u=\frac{z\bar{z}+ \frac{1}{2}z^2\barζ+\frac{1}{2} \bar{z}^2 ζ}{1-ζ\barζ}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone $\{x_1^2+x_2^2-x_3^2=0\}$. Another representation is the tube $u=\frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: \[ (z,ζ,w) \longmapsto \big( f(z,ζ), g(z,ζ), ρ\,w+h(z,ζ) \big) =: (z',ζ',w'). \] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiola's formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\bar{z}+\frac{1}{2}\,z^2\barζ +\frac{1}{2}\,\bar{z}^2ζ}{ 1-ζ\barζ} + \sum_{a,b,c,d \atop a+c\geqslant 3}\, G_{a,b,c,d}\, z^aζ^b\bar{z}^c\barζ^d, \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {\rm Im}\, G_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0, 2, 3, 0}$, ${\rm Re} G_{3,0,1,1}$. With this, a brige Poincaré $\longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.