  The spectrum of a hexagonal ring is analysed using concepts of group theory
and a tight-binding model with first, second and third neighbours. The two
doublets in the spectrum are explained with the $C_3$ symmetry group together
with time-reversal symmetry. Degeneracy lifts are induced by means of various
mechanisms. Conjugation symmetry breaking is introduced via magnetic fields,
while $C_3$ breaking is studied with the introduction of defects, similar to
the inclusion of fluorine atoms. Concrete applications to benzene and borazine
are shown to illustrate the generality of our description. Wave functions are
described in connection with partial or full aromaticity. Electronic density
currents are found for all cases. A detailed study of a supersymmetry in a
$6$-ring is presented and its consequences on electronic spectra are discussed.
