In this talk, I will construct a moduli space of generalized hyperpolygons from a comet-shaped quiver. The resulting Nakajima quiver variety can be interpreted as a distinguished subvariety of a moduli space of meromorphic Higgs bundles on a punctured curve. I will discuss how the moduli space inherits, for complete and minimal flags, a Gelfand-Tsetlin-type integrable system from the reduction of a product of cotangent bundles of (partial) flag varieties. This is joint work with Laura Schaposnik.