XXZ spin chain ($H=J{\sum }_j(S_j^x{S}_{j+1}^x+S_j^y{S}_{j+1}^y+\Delta S_j^z{S}_{j+1}^z)$)

The XXZ spin chain has equal transverse exchange interactions and a possibly different longitudinal coupling. For $J>0$ the transverse coupling is antiferromagnetic, while the sign and size of $\Delta$ determine the longitudinal preference. A periodic even chain is bipartite. Its continuous symmetry is rotation about $z$ for generic $\Delta$, enhanced to full spin rotation at $\Delta =1$.

For an even periodic classical chain with spin length $S$ and $J>0$, each bond attains the negative largest singular value of $\mathrm{diag}(1,1,\Delta )$. For $\Delta >1$ the two minimizing configurations are alternating $z$ spins. For $|\Delta |<1$ they are planar Néel states with arbitrary in-plane angle. For $\Delta <-1$ the two configurations have uniform $z$ spins. At $\Delta =1$ the Néel direction is arbitrary on a sphere; at $\Delta =-1$ a staggered transverse component and uniform longitudinal component parameterize another spherical degeneracy. These are classical configuration manifolds, not a claim about finite quantum-chain degeneracy.