  Pfaff lattice was introduced by Adler and van Moerbeke to describe the
partition functions for the random matrix models of GOE and GSE type. The
partition functions of those matrix models are given by the Pfaffians of
certain skew-symmetric matrices called the moment matrices, and they are the
$\tau$-functions of the Pfaff lattice. In this paper, we study a finite version
of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove
the complete integrability in the sense of Arnold-Liouville, and using a moment
map, we describe the real isospectral varieties of the Pfaff lattice. The image
of the moment map is a convex polytope whose vertices are identified as the
fixed points of the flow generated by the Pfaff lattice.
