Lie groups, cluster variables and integrable systems

We discuss Poisson structures on Lie groups and propose an explicit construction of integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to co-extended loop groups, not only gives rise to several alternative descriptions of relativistic Toda systems but also allows to formulate in general terms some new class of integrable models. We discuss the subtleties of this Lax map in relation to the ambiguity in projection to the trivial co-extension and propose a way to write the spectral curve equation, which fixes this ambiguity, for the periodic Toda chain and its generalisations.