  The dynamics of waves in periodic media is determined by the band structure
of the underlying periodic Hamiltonian. Symmetries of the Hamiltonian can give
rise to novel properties of the band structure. Here we consider a class of
periodic Schr\"odinger operators, $H_V=-\Delta+V$, where $V$ is periodic with
respect to the lattice of translates $\Lambda=\mathbb{Z}^2$. The potential is
also assumed to be real-valued, sufficiently regular and such that, with
respect to some origin of coordinates, inversion symmetric (even) and invariant
under $\pi/2$ rotation. The present results are the $\mathbb{Z}^2-$ analogue of
results obtained for conical degenerate points (Dirac points) in honeycomb
structures. Our proofs make use of the framework developed by
Fefferman-Weinstein and Fefferman-Lee-Thorp-Weinstein.
