We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}$ for a ring of $N$ spin-1/2 particles with associated spin vector operator $(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}$ for the $j$th spin. Here ${r}_{j,k}$ is the chord distance between sites $j$ and $k$. The case $\ensuremath{\alpha}=2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\ensuremath{\alpha}\ensuremath{\ne}2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\ensuremath{\alpha}=2$ from $\ensuremath{\alpha}\ensuremath{\ne}2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\ensuremath{\alpha}=2$. Whereas for $\ensuremath{\alpha}\ensuremath{\ne}2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$.