  In realizing practical non-trivial topological electronic phases stable
structures need to be determined first. Tin and lead do stabilize an optimal
two-dimensional high-buckled phase --a hexagonal-close packed bilayer structure
with nine-fold atomic coordination-- and they do not stabilize topological
fullerenes, as demonstrated by energetics, phonon dispersion curves, and the
structural optimization of finite-size samples. The high-buckled phases are
metallic due to their high atomic coordination. The optimal structure of
fluorinated tin lacks three-fold symmetry and it stabilizes small samples too.
It develops two oblate conical valleys on the first Brillouin zone coupling
valley, sublattice, and spin degrees of freedom with a novel
$\tau_z\sigma_xs_x$ term, thus making it a new 2D platform for valleytronics.
