  Given the Hermitian, symmetric and symplectic ensembles, it is shown that the
probability that the spectrum belongs to one or several intervals satisfies a
nonlinear PDE. This is done for the three classical ensembles: Gaussian,
Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary
points of the intervals) is related to the Toda lattice and the KP equation,
whereas for the symmetric and symplectic ensembles the PDE is an inductive
equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The
method consists of inserting time-variables in the integral and showing that
this integral satisfies integrable lattice equations and Virasoro constraints.
