On the Complete Integrability of some Lax Equations on a Periodic Lattice

We consider some Lax equations on a periodic lattice with $N=2$ sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.