  We study the formation of bound states and three-component bright vector
solitons in a quasi-one-dimensional spin-orbit-coupled hyperfine spin $f=1$
Bose-Einstein condensate using numerical solution and variational approximation
of a mean-field model. In the antiferromagnetic domain, the solutions are
time-reversal symmetric, and the component densities have multi-peak structure.
In the ferromagnetic domain, the solutions violate time-reversal symmetry, and
the component densities have single-peak structure. The dynamics of the system
is not Galelian invariant. From an analysis of Galelian invariance, we
establish that the single-peak ferromagnetic vector solitons are true solitons
and can move maintaining constant component densities, whereas the
antiferromagnetic solitons cannot move with constant component densities.
