  While different ways to realize spin-orbit coupling in Bose-Einstein
condensates exist, not all are currently experimentally implementable. Here we
present a detailed study of gap solitons in a Bose- Einstein condensate with
experimentally realizable spin-orbit coupling and discuss two cases relating to
a spin-dependent parity symmetry. In the parity symmetric case, two families of
fundamental gap solitons in second linear energy gap are demonstrated with
opposite sign of the parity, with one family having single humped densities and
the other double humped ones. In the case of broken parity symmetry, the
fundamental modes manifest spin-polarization. Both families possess an opposite
sign of the spin-polarization.
