  Fels-Kaup (Acta Mathematica 2008) classified homogeneous $\mathfrak{C}_{2,1}$
hypersurfaces $M^5 \subset \mathbb{C}^3$ and discovered that they are all
biholomorphic to tubes $S^2 \times i \mathbb{R}^3$ over some affinely
homogeneous surface $S^2 \subset \mathbb{R}^3$. The second and third authors in
2003.08166, by performing highly non-straightforward calculations, conducted
the Cartan method of equivalence to classify homogeneous models of PDE systems
related to such $\mathfrak{C}^{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$.
Kolar-Kossovskiy 1905.05629 and the authors 2003.01952 constructed a formal and
a convergent Poincar\'e-Moser normal form for $\mathfrak{C}_{2,1}$
hypersurfaces $M^5 \subset \mathbb{C}^3$. But this was only a first,
preliminary step. Indeed, the invariant branching tree underlying Fels-Kaup's
classification was still missing in the literature, due to computational
obstacles. The present work applies the power series method of equivalence,
confirms Fels-Kaup 2008, and finds a differential-invariant tree. To terminate
the middle (thickest) branch, it is necessary to compute up to order $10$ with
$5$ variables. Again, calculations, done by hand, are non-straightforward.
