In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector $k$-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.
Helminck, G. F., van de Leur, J. W., & van de Leur, J. W. (1998). An analytic description of the vector constrained KP hierarchy. Communications in mathematical physics, 193(3), 627-641. https://doi.org/10.1007/s002200050341