Defining equations of $7$-dimensional model CR hypersurfaces

We study CR hypersurfaces in $\mathbb{C}^4$ that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^4$, and present a detailed study of their local invariants. The normal form illustrates that $2$-nondegenerate models in $\mathbb{C}^4$ comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of $2$-nondegenerate structures in $\mathbb{C}^3$. In further contrast to these previously studied settings, we demonstrate that not all $2$-nondegenerate structures in $\mathbb{C}^4$ arise as perturbations of homogeneous models. We derive defining equations for the homogeneous $2$-nondegenerate models, a set of $9$ structures, and find explicit formulas for their infinitesimal symmetries.