Inverse spectral problem for GK integrable system

Abstract

A.Goncharov and R.Kenyon has defined a class of integrable system on a cluster varieties constructed out of a Newton polygon on the plane. In the present note we show that thiest cluster varieties coincides with the configuration spaces of collections of complete flags in an infinite dimensional space invariant with respect to two commuting operators. We use this interpretation to give explicit solution for these integrable systems in terms of theta-functions.