Integrals of motion of the classical lattice sine-Gordon system

We compute the local integrals of motions of the classical limit of the lattice sine-Gordon system, using a geometrical interpretation of the local sine-Gordon variables. Using an analogous description of the screened local variables, we show that these integrals are in involution. We present some remarks on relations with the situation at the roots of 1 and results on another latticization (linked to the principal subalgebra of rather than the homogeneous one). Finally, we analyze a module of "screened semilocal variables," on which the whole acts.