The Pfaff lattice on symplectic matrices

The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [{\it Intern. Math. Res. Notices}, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, {\it Physica D}, {\bf 91} (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by $2\times 2$ block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.