  In this short note we provide two extensions on the recent explicit results
on the matrix-product ansatz for the non-equilibrium steady state of a
markovianly boundary-driven anisotropic Heisenberg XXZ spin 1/2 chain. We write
a perturbative solution for the steady state density matrix in the system-batyh
coupling for an arbitrary (asymmetric) set of four spin-flip rates at the two
chain ends, generalizing the symmetric-driving ansatz of [Phys. Rev. Lett. 106,
217206 (2011)]. Furthermore, we generalize the exact (non-perturbative) form of
the steady state for just two Lindblad channels (spin-up flipping on the left,
and spin-down flipping on the right) to an arbitrary (asymmetric) ratio of the
spin flipping rates [Phys. Rev. Lett. 107, 137201 (2011)]. In addition, we also
indicate a simple proof of uniqueness of our steady states.
