Rational solutions to the Pfaff lattice and Jack polynomials

The finite Pfaff lattice is given by a commuting Lax pair involving, a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t1, t2,..., after conjugation by a diagonal matrix. The sequence of polynomial tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of tau-functions is given inductively by the action of a fixed vertex operator. As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.