Affine Homogeneity of Indefinite Real Hypersurfaces in the Space $\mathbb{C}^3$

We develop a constructive approach to the problem of describing affinely homogeneous real hypersurfaces in $3$-dimensional complex space having nondegenerate sign-indefinite Levi form. We construct the affine invariants of a nondegenerate indefinite hypersurface in terms of second-order jets of its defining function and introduce the notion of the affine canonical equation of this surface. Three main types of canonical equations are considered. For each of these types, we construct a family of Lie algebras related to affinely homogeneous surfaces of a particular type. As a result, a family (depending on two real parameters) of affinely different homogeneous submanifolds of $3$-dimensional complex space is presented (as matrix algebras).