  We consider a two-dimensional periodic Schr\"odinger operator $H=-\Delta+W$
with $\Gamma$ being the lattice of periods. We investigate the structure of the
edges of open gaps in the spectrum of $H$. We show that under arbitrary small
perturbation $V$ periodic with respect to $N\Gamma$ where $N=N(W)$ is some
integer, all edges of the gaps in the spectrum of $H+V$ which are perturbation
of the gaps of $H$ become non-degenerate, i.e. are attained at finitely many
points by one band function only and have non-degenerate quadratic
minimum/maximum. We also discuss this problem in the discrete setting and show
that changing the lattice of periods may indeed be unavoidable to achieve the
non-degeneracy.
