  We consider the two-spin subsystem entanglement for eigenstates of the
Hamiltonian
  \[ H= \sum_{1\leq j< k \leq N} (\frac{1}{r_{j,k}})^{\alpha} {\mathbf
\sigma}_j\cdot {\mathbf \sigma}_k \] for a ring of $N$ spins 1/2 with
asssociated spin vector operator $(\hbar /2){\bf \sigma}_j$ for the $j$-th
spin. Here $r_{j,k}$ is the chord-distance betwen sites $j$ and $k$. The case
$\alpha =2$ corresponds to the solvable Haldane-Shastry model whose spectrum
has very high degeneracies not present for $\alpha \neq 2$. Two spin subsystem
entanglement shows high sensistivity and distinguishes $\alpha =2$ from $\alpha
\neq 2$. There is no entanglement beyond nearest neighbors for all eigenstates
when $\alpha =2$. Whereas for $\alpha \neq 2$ one has selective entanglement at
any distance for eigenstates of sufficiently high energy in a certain interval
of $\alpha$ which depends on the energy. The ground state (which is a singlet
only for even $N$) does not have entanglement beyond nearest neighbors, and the
nearest neighbor entanglement is virtually independent of the range of the
interaction controlled by $\alpha$.
