  We construct exact steady states of unitary non-equilibrium time evolution in
the gapless XXZ spin-1/2 chain where integrability preserves ballistic spin
transport at long-times. We characterize the quasi-local conserved quantities
responsible for this feature and introduce a computationally effective way to
evaluate their expectation values on generic matrix product initial states. We
employ this approach to reproduce the long-time limit of local observables in
all quantum quenches which explicitly break particle-hole or time-reversal
symmetry. We focus on a class of initial states supporting persistent spin
currents and our predictions remarkably agree with numerical simulations at
long times. Furthermore, we propose a protocol for this model where
interactions, even when antiferromagnetic, are responsible for the unbounded
growth of a macroscopic magnetic domain.
