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In this paper, we prove that in the system of bending flows on the modulispace of polygons with fixed side lengths introduced by Kapovich and Millson,the singular fibers are isotropic homogeneous submanifolds. The proof coversthe case where the system is defined by any maximal family of disjointdiagonals. We also take in account the case where the fixed side lengths arenot generic. In this case, the phase space is an orbispace, and our resultholds in the sense that singular fibers are isotropic orbispaces. In a lastpart we provide leads in favor of a similar study of the integrable systemsdefined by Nohara and Ueda on the Grassmannian of 2-planes in $\mathbb{C}^n$.

Singular fibers of the bending flows on the moduli space of 3D polygons