  Degeneracy is an omnipresent phenomenon in various physical systems, which
has its roots in the preservation of geometrical symmetry. In electronic and
photonic crystal systems, very often this degeneracy can be broken by virtue of
strong interactions between photonic modes of the same energy, where the level
repulsion and the hybridization between modes causes the emergence of photonic
bandgaps. However, most often this phenomenon does not lead to a complete and
inverted bandgap formation over the entire Brillouin zone. Here, by
systematically breaking the symmetry of a two-dimensional square photonic
crystal, we investigate the formation of Dirac points, line node singularities,
and inverted bandgaps. The formation of this complete bandgap is due to the
level repulsion between degenerate modes along the line nodes of a
semimetal-like photonic crystal, over the entire Brillouin zone. Our numerical
experimentations are performed by a home-build numerical framework based on a
multigrid finite element method. The developed numerical toolbox and our
observations pave the way towards designing complete bandgap photonic crystals
and exploring the role of symmetry on the optical behaviour of even more
complicated orders in photonic crystal systems.
