  We describe a class of integrable systems on Poisson submanifolds of the
affine Poisson-Lie groups $\widehat{PGL}(N)$, which can be enumerated by
cyclically irreducible elements the co-extended affine Weyl groups
$(\widehat{W}\times \widehat{W})^\sharp$. Their phase spaces admit cluster
coordinates, whereas the integrals of motion are cluster functions. We show,
that this class of integrable systems coincides with the constructed by
Goncharov and Kenyon out of dimer models on a two-dimensional torus and
classified by the Newton polygons. We construct the correspondence between the
Weyl group elements and polygons, demonstrating that each particular integrable
model admits infinitely many realisations on the Poisson-Lie groups. We also
discuss the particular examples, including the relativistic Toda chains and the
Schwartz-Ovsienko-Tabachnikov pentagram map.
