  It is known that the interplay of the spin-orbit-coupling (SOC) and
mean-field self-attraction creates stable two-dimensional (2D) solitons (ground
states) in spinor Bose-Einstein condensates. However, SOC destroys the system's
Galilean invariance, therefore moving solitons exist only in a narrow interval
of velocities, outside of which the solitons suffer delocalization. We
demonstrate that the application of a relatively weak moving optical lattice
(OL), with the 2D or quasi-1D structure, makes it possible to greatly expand
the velocity interval for stable motion of the solitons. The stability domain
in the system's parameter space is identified by means of numerical methods. In
particular, the quasi-1D OL produces a stronger stabilizing effect than its
full 2D counterpart. Some features of the domain are explained analytically.
