  In this paper, we prove a polynomial Central Limit Theorem for several
integrable models, and for the $\beta$-ensembles at high-temperature with
polynomial potential. Furthermore, we connect the mean values, the variances
and the correlations of the moments of the Lax matrices of these integrable
systems with the ones of the $\beta$-ensembles. Moreover, we show that the
local functions' space-correlations decay exponentially fast for the considered
integrable systems. For these models, we also established a Berry--Esseen type
bound.
