  $2$-nondegenerate real hypersurfaces in complex manifolds play an important
role in CR-geometry and the theory of Hermitian Symmetric Domains. In this
paper, we construct a complete convergent normal form for everywhere
$2$-nondegenerate real-analytic hypersurfaces in complex $3$-space. We do so by
developing the homological approach of Chern-Moser in the $2$-nondegenerate
setting. This seems to be the first such construction for hypersurfaces of
infinite Catlin multitype. Our approach is based on using a rational
(nonpolynomial) model for everywhere $2$-nondegenerate hypersurfaces, which is
the local realization due to Fels-Kaup of the well known tube over the light
cone.
  As an application, we obtain, in the spirit of Chern-Moser theory, a
criterion for the local sphericity (i.e. local equivalence to the model) for a
$2$-nondegenerate hypersurface in terms of its normal form. As another
application, we obtain an explicit description of the moduli space of
everywhere $2$-nondegenerate hypersurfaces.
