Integrable Systems and Rank-One Conditions for Rectangular Matrices

We give a determinantal formula for tau functions of the KP hierarchy in terms of rectangular constant matrices $A$, $B$, and $C$ satisfying a rank-one condition. This result is shown to generalize and unify many previous results of different authors on constructions of tau functions for differential and difference integrable systems from square matrices satisfying rank-one conditions. In particular, its explicit special cases include Wilson's formula for tau functions of the rational KP solutions in terms of Calogero–Moser Lax matrices and our previous formula for the KP tau functions in terms of almost-intertwining matrices.