  We examine the origin of the formation of narrow bands in LK-99
(Pb$_{9}$Cu(PO$_4$)$_6$O) and the parent compound without the Cu doping using
density functional theory calculations and model Hamiltonian studies. Explicit
analytical expressions are given for a nearest-neighbor tight-binding (TB)
Hamiltonian in the momentum space for both the parent and the LK-99 compound,
which can serve as an effective model to study various quantum phenomena
including superconductivity. The parent material is an insulator with the
buckle oxygen atom on the stacked triangular lattice forming the topmost bands,
well-separated from the remaining oxygen band manifold. The $C_3$
symmetry-driven two-band TB model describes these two bands quite well. These
bands survive in the Cu-doped, LK-99, though with drastically altered band
dispersion due to the Cu-O interaction. A similar two-band model involving the
Cu $xz$ and $yz$ orbitals broadly describes the top two valence bands of LK-99.
However, the band dispersions of both the Cu and O bands are much better
described by the four-band TB model incorporating the Cu-O interactions on the
buckled honeycomb lattice. We comment on the possible mechanisms of
superconductivity in LK-99. even though the actual T$_c$ may be much smaller
than reported, and suggest that interstitial Cu clusters leading to broad bands
might have a role to play
