  Precise asymptotics known for the Green function of the Laplacian have found
their analogs for bounded below periodic elliptic operators of the second-order
below and at the bottom of the spectrum. Due to the band-gap structure of the
spectra of such operators, the question arises whether similar results can be
obtained near or at the edges of spectral gaps. In a previous work, two of the
authors considered the case of a spectral edge. The main result of this article
is finding such asymptotics near a gap edge, for "generic" periodic elliptic
operators of second-order with real coefficients in dimension $d \geq 2$, when
the gap edge occurs at a symmetry point of the Brillouin zone.
