  We study the bipartite von Neumann entanglement entropy and matrix elements
of local operators in the eigenstates of an interacting integrable Hamiltonian
(the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that
of quantum chaotic systems. We find that the leading term of the average (over
all eigenstates in the zero magnetization sector) eigenstate entanglement
entropy has a volume-law coefficient that is smaller than the universal
(maximal entanglement) one in quantum chaotic systems. This establishes the
entanglement entropy as a powerful measure to distinguish integrable models
from generic ones. Remarkably, our numerical results suggest that the
volume-law coefficient of the average entanglement entropy of eigenstates of
the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for
translationally invariant quadratic fermionic models. We also study matrix
elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian
at the center of the spectrum. For the diagonal matrix elements, we show
evidence that the support does not vanish with increasing system size, while
the average eigenstate-to-eigenstate fluctuations vanish in a power-law
fashion. For the off-diagonal matrix elements, we show that they follow a
distribution that is close to (but not quite) log-normal, and that their
variance is a well-defined function of $\omega=E_{\alpha}-E_{\beta}$
($\{E_{\alpha}\}$ are the eigenenergies) proportional to $1/D$, where $D$ is
the Hilbert space dimension.
