A large class of examples of integrable systems arise on certain submanifolds of affine Lie groups. Such systems are enumerated out of certain elements of affine Weyl groups. The most commonly known integrable system of this class is the relativistic Toda chain, but many other ones, like integrable tops, fit into this class. We shall present another construction of integrable systems due to A.Goncharov and R.Kenyon, who construct them constructed out of integral convex polygons on the plane. These two classes of integrable systems turn out to be almost the same. However, Goncharov-Kenyon point of view gives a new insight on the whole world of integrable systems (and thus can be used as an introduction to the subject). For example, the phase space in the GK approach comes equipped with a cluster variety structure, which implies that it has simple parametrization, has a discrete group action, has a canonical basis of function on it, can be easily quantized and has other nice properties. Another advantage of the GK approach is that the construction of integrable system uses a new technique to the subject, namely it is given by dimer partition functions.