Normal forms for rigid c_2,1 hypersurfaces m⁵ ⊂ c³

Consider a 2-nondegenerate constant Levi rank 1 rigid C^ω hypersurface M⁵ ⊂ C³ in coordinates (z, ζ, w = u + iv): u = F (z, ζ, z, ζ). The Gaussier-Merker model u = [Formula Presented] was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone {x²1 +x²2 −x²3 = 0}. Another representation is the tube u =[Formula Presented] . 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 goal is to establish the Poincaré-Moser complete normal form: u =[Formula Presented] with 0 = G_a,b,0,0 = G_a,b,1,0 = G_a,b,2,0 and 0 = G_3,0,0,1 = Im G_3,0,1,1 .