  The repartition of the separation between energy levels of various isotropic
S=1/2 antiferromagnetic chains is studied numerically with the aim of
investigating the transition from integrable to non-integrable systems. We
begin by displaying the level separation distribution of the integrable Bethe
chain. Then two non-integrable systems, two coupled chains and a
next-nearest-neighbor coupled chain, are studied as a function of the coupling.
We examine how the level spacing evolves from the Poisson distribution to the
GOE distribution. Finally we consider the Haldane-Shastry $1/r^{2}$ model. A
number of conclusions regarding the behaviour and relevance of the level
spacing distribution in these spin systems is drawn.
