  We study relations between the eigenvectors of rational matrix functions on
the Riemann sphere. Our main result is that for a subclass of functions that
are products of two elementary blocks it is possible to represent these
relations in a combinatorial-geometric way using a diagram of a cube. In this
representation, vertices of the cube represent eigenvectors, edges are labeled
by differences of locations of zeroes and poles of the determinant of our
matrix function, and each face corresponds to a particular choice of a
coordinate system on the space of such functions. Moreover, for each face this
labeling encodes, in a neat and efficient way, a generating function for the
expressions of the remaining four eigenvectors that label the opposing face of
the cube in terms of the coordinates represented by the chosen face. The main
motivation behind this work is that when our matrix is a Lax matrix of a
discrete integrable system, such generating functions can be interpreted as
Lagrangians of the system, and a choice of a particular face corresponds to a
choice of the direction of the motion.

Exactly Solvable and Integrable Systems