  It was recently found that, under the action of the spin-orbit coupling (SOC)
and Zeeman splitting (ZS), binary BEC with intrinsic cubic nonlinearity
supports families of gap solitons, provided that the kinetic energy is
negligible in comparison with the SOC and ZS terms. We demonstrate that, also
under the action of SOC and ZS, a similar setting may be introduced for BEC
with two components representing different atomic states, resonantly coupled by
microwave radiation, while the Poisson equation accounts for the feedback of
the two-component atomic wave function onto the radiation. The
microwave-mediated interaction induces an effective nonlinear trapping
potential, which strongly affects the purport of the linear spectrum in this
system. As a result, families of both gap and embedded solitons (those
overlapping with the continuous spectrum) are found, being chiefly stable. The
shape of the solitons features exact or broken skew symmetry. In addition to
fundamental solitons (whose shape may or may not include a node), a family of
dipole solitons is constructed too, which are even more stable than their
fundamental counterparts. A nontrivial stability area is identified for moving
solitons in the present system, which lacks Galilean invariance. Colliding
solitons merge into a single one.
