  Five-component minimum-energy bound states and mobile vector solitons of a
spin-orbit-coupled quasi-one-dimensional hyperfine-spin-2 Bose-Einstein
condensate are studied using the numerical solution and variational
approximation of a mean-field model. Two distinct types of solutions with
single-peak and multi-peak density distribution of the components are
identified in different domains of interaction parameters. From an analysis of
Galilean invariance and time-reversal symmetry of the Hamiltonian, we establish
that vector solitons with multi-peak density distribution preserve
time-reversal symmetry, but cannot propagate maintaining the shape of
individual components. However, those with single-peak density distribution
violate time-reversal symmetry of the Hamiltonian, but can propagate with a
constant velocity maintaining the shape of individual components.
