studied the magnetism of the ultrathin wires of fifth and sixth row elements supported on Cu(117) and Ag(117) vicinal surfaces. They found that only metals with a half-filled $d$ band are found to have magnetic order on Cu(117) surface, additionally the ferromagnetic order is energetically stable. On the contrary, on the Ag(117) surface, the anti-ferromagnetic order is stable. The magnetism of mono-atomic chain is strongly dependent on their local environment absorbed on the surfaces.

Besides the high-index vicinal surfaces such as Ag(117), the reconstructed Au(110)-(1$\times$2) surface decorated with one-dimensional troughs along the closed-packed A-\[1$\bar{1}$0\] direction can be used as a template to grow one-dimensional nano-structure. Experimentally, the distribution of lengths of Ni chains shows that most Ni chains are short than 6 [@Hitzke1]. Our molecular simulation additionally shows that the lattice misfit between Ni chains and the Au substrate can induce the instability of Ni chains and makes them break into short segments [@Fan1].

In this work we study the magnetism of mono-atomic Ni chains supported on Au(110)-(1$\times$2) surface using the same theoretical method as in Ref. [@Spisak2]. Our studies focus on the magnetic anisotropy of surface-supported magnetic atomic chains. We perform the non-collinear magnetic calculations including the spin-orbit coupling 