  Existence and stability of Dirac points in the dispersion relation of
operators periodic with respect to the hexagonal lattice is investigated for
different sets of additional symmetries. The following symmetries are
considered: rotation by $2\pi/3$ and inversion, rotation by $2\pi/3$ and
horizontal reflection, inversion or reflection with weakly broken rotation
symmetry, and the case where no Dirac points arise: rotation by $2\pi/3$ and
vertical reflection.
  All proofs are based on symmetry considerations. In particular, existence of
degeneracies in the spectrum is deduced from the (co)representation of the
relevant symmetry group. The conical shape of the dispersion relation is
obtained from its invariance under rotation by $2\pi/3$. Persistence of conical
points when the rotation symmetry is weakly broken is proved using a geometric
phase in one case and parity of the eigenfunctions in the other.
