  We demonstrate stable and metastable vortex-bright solitons in a
three-dimensional spin-orbit-coupled three-component hyperfine spin-1
Bose-Einstein condensate (BEC) using numerical solution and variational
approximation of a mean-field model. The spin-orbit coupling provides
attraction to form vortex-bright solitons in both attractive and repulsive
spinor BECs. The ground state of these vortex-bright solitons is axially
symmetric for weak polar interaction. For a sufficiently strong ferromagnetic
interaction, we observe the emergence of a fully asymmetric vortex-bright
soliton as the ground state. We also numerically investigate moving solitons.
The present mean-field model is not Galilean invariant, and we use a
Galilean-transformed mean-field model for generating the moving solitons.
