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We investigate a class of CR invariants of uniformly $2$-nondegenerate CRhypersurfaces, termed modified symbols, and develop methods for derivingdefining equations of CR hypersurfaces realizing different values of theseinvariants. For hypersurfaces in $\mathbb{C}^4$, we obtain the fullclassification of modified symbols realizable by weighted homogeneous (withrespect to a natural weighting system) rigid CR hypersurfaces, and obtaindefining equations for the recently classified $7$-dimensional homogeneousmodified symbol models. Moreover, we characterize, up to CR equivalence, allweighted homogeneous rigid uniformly $2$-nondegenerate CR hypersurfaces in$\mathbb{C}^{N}$ whose holomorphic infinitesimal symmetries generate atransitive action on their Levi leaf spaces.

Defining equations of $7$-dimensional model CR hypersurfaces and models in $\mathbb{C}^{N}$