  We propose new classes of random matrix ensembles whose statistical
properties are intermediate between statistics of Wigner-Dyson random matrices
and Poisson statistics. The construction is based on integrable N-body
classical systems with a random distribution of momenta and coordinates of the
particles. The Lax matrices of these systems yield random matrix ensembles
whose joint distribution of eigenvalues can be calculated analytically thanks
to integrability of the underlying system. Formulas for spacing distributions
and level compressibility are obtained for various instances of such ensembles.
