  We discuss the Poisson structures on Lie groups and propose an explicit
construction of the integrable models on their appropriate Poisson
submanifolds. The integrals of motion for the SL(N)-series are computed in
cluster variables via the Lax map. This construction, when generalised to the
co-extended loop groups, gives rise not only to several alternative
descriptions of relativistic Toda systems, but allows to formulate in general
terms some new class of integrable models.
