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

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}z^2bar{zeta}+frac{1}{2} bar{z}^2 zeta}{1-zeta bar{zeta}}$ 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,zeta,w) longmapsto big( f(z,zeta), g(z,zeta), rho,w+h(z,zeta) big) =: (z,zeta,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 Pocchiolas formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincare-Moser complete normal form: [ u = frac{zbar{z}+frac{1}{2},z^2bar{zeta} +frac{1}{2},bar{z}^2zeta}{ 1-zetabar{zeta}} + sum_{a,b,c,d atop a+cgeqslant 3}, G_{a,b,c,d}, z^azeta^bbar{z}^cbar{zeta}^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 Poincare $longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.