Integrable Systems and Rank One Conditions for Rectangular Matrices

We provide 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, it contains as explicit special cases the formula of Wilson for tau-functions of rational KP solutions in terms of Calogero-Moser Lax matrices as well as our previous formula for KP tau functions in terms of almost-intertwining matrices.