  Beta-ensembles of random matrices are naturally considered as quantum
integrable systems, in particular, due to their relation with conformal field
theory, and more recently appeared connection with quantized Painlev\'e
Hamiltonians. Here we demonstrate that, at least for {\it even integer} beta,
these systems are classically integrable, e.g. there are Lax pairs associated
with them, which we explicitly construct. To come to the result, we show that a
solution of every Fokker-Planck equation in one space (and one time) dimensions
can be considered as a component of an eigenvector of a Lax pair. The explicit
finding of the Lax pair depends on finding a solution of a governing system --
a closed system of two nonlinear PDEs of hydrodynamic type. This result
suggests that there must be a solution for all values of beta. We find the
solution of this system for even integer beta in the particular case of quantum
Painlev\'e II related to the soft edge of the spectrum for beta-ensembles. The
solution is given in terms of Calogero system of $\beta/2$ particles in an
additional time-dependent potential. Thus, we find another situation where
quantum integrability is reduced to classical integrability.
