Classification of Affinely Homogeneous Hessian Rank 2 Hypersurfaces S³ in R⁴

We determine all affinely homogeneous hypersurfaces S³ in R⁴ whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of homogeneous models of hypersurfaces S³ in R⁴, sometimes parametrized by a certain complicated algebraic variety, especially for the 15 (over 34) families of models which are simply transitive. We employ the power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. In Lie's original classification spirit, we describe the found homogeneous models by listing explicit Lie algebras of infinitesimal transformations, sometimes parametrized by absolute invariants satisfying certain algebraic equations.