  We address the old and widely debated question of the statistical properties
of integrable quantum systems, through the analysis of the paradigmatic
Lieb-Liniger model. This quantum many-body model of 1-d interacting bosons
allows for the rigorous determination of energy spectra via the Bethe ansatz
approach and our interest is understanding whether Poisson statistics is a
characteristic feature of this model. Using both analytical and numerical
studies we show that the properties of spectra strongly depend on whether the
analysis is done for a full energy spectrum or for a single subset with fixed
total momentum. We show that the Poisson distribution of spacing between
nearest-neighbor energies can occur only for a set of energy levels with fixed
total momentum, for neither too large nor too weak interaction strength, and
for sufficiently high energy. On the other hand, when studying long-range
correlations between energy levels, we found strong deviations from the
predictions given by a Poisson process.
