  The aim of this paper is studying from an alternative point of view the
integrability of the spin chain with long-range elliptic interactions
introduced by Inozemtsev. Our analysis relies on some well-established
conjectures characterizing the chaotic vs. integrable behavior of a quantum
system, formulated in terms of statistical properties of its spectrum. More
precisely, we study the distribution of consecutive levels of the (unfolded)
spectrum, the power spectrum of the spectral fluctuations, the average
degeneracy, and the equivalence to a classical vertex model. Our results are
consistent with the general consensus that this model is integrable, and that
it is closer in this respect to the Heisenberg chain than to its trigonometric
limit (the Haldane-Shastry chain). On the other hand, we present some numerical
and analytical evidence showing that the level density of Inozemtsev's chain is
asymptotically Gaussian as the number of spins tends to infinity, as is the
case with the Haldane-Shastry chain. We are also able to compute analytically
the mean and the standard deviation of the spectrum, showing that their
asymptotic behavior coincides with that of the Haldane-Shastry chain.
