Models of $2$-nondegenerate CR hypersurface in ${\mathbb{C}}^N$

We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics do in the Levi nondegenerate case. We characterize all $2$-nondegenerate models and show that the moduli space of such hypersurfaces in ${\mathbb{C}}^N$ is infinite dimensional for each $N>3$. We derive a normal form for these models’ defining equations that is unique up to an action of a finite dimensional Lie group. We generalize recently introduced CR invariants termed modified symbols, and show how to compute these intrinsically defined invariants from a model’s defining equation. We show that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel and derive explicit formulas for them.