  Consider a symmetric (finite) matrix ensemble, with a certain probability
distribution. What is the probability that the spectrum belongs to a certain
interval or union of intervals on the real line? In this paper, we show that,
upon introducing an appropriate time parameter, this probability is intimately
related to Pfaffians, which as a vector satisfy the so-called Pfaff lattice.
The latter is a particular reduction of the 2d-Toda lattice. In particular,
they satisfy a KP-like equation, but with a right hand side, depending on
nearest neighbors. They also satisfy Virasoro constraints, which combined with
the KP-like equation lead to inductive equations for the probabilities.
