  We study the average and the standard deviation of the entanglement entropy
of highly excited eigenstates of the integrable interacting spin-$\frac{1}{2}$
XYZ chain away from and at special lines with $U(1)$ symmetry and
supersymmetry. We universally find that the average eigenstate entanglement
entropy exhibits a volume-law coefficient that is smaller than that of
quantum-chaotic interacting models. At the supersymmetric point, we resolve the
effect that degeneracies have on the computed averages. We further find that
the normalized standard deviation of the eigenstate entanglement entropy decays
polynomially with increasing system size, which we contrast to the exponential
decay in quantum-chaotic interacting models. Our results provide state-of-the
art numerical evidence that integrability in spin-$\frac{1}{2}$ chains reduces
the average, and increases the standard deviation, of the entanglement entropy
of highly excited energy eigenstates when compared to those in quantum-chaotic
interacting models.
