  We construct a family of integrable Hamiltonian systems generalizing the
relativistic periodic Toda lattice, which is recovered as a special case. The
phase spaces of these systems are double Bruhat cells corresponding to pairs of
Coxeter elements in the affine Weyl group. In the process we extend various
results on double Bruhat cells in simple algebraic groups to the setting of
Kac-Moody groups. We also generalize some fundamental results in Poisson-Lie
theory to the setting of ind-algebraic groups, which is of interest beyond our
immediate applications to integrable systems.
