We introduce a completely integrable system on the Grassmannian of 2-planes in an -space associated with any triangulation of a polygon with sides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.
"Toric degenerations of integrable systems on Grassmannians and polygon spaces." Nagoya Math. J. 214 125 - 168, June 2014. https://doi.org/10.1215/00277630-2643839