Compatibility between the Lax, Hamilton and St. Petersburg formalisms in terms of the conserved quantities.

Question

In classical integrable models, in the discrete case we have the Lax equation discrete $\frac{dL_{n}}{dt} = M_{n+1}L_{n}-L_{n}M_{n}$ the Sklyanin algebra, $\lbrace T_{a}(u),T_{b}(v)\rbrace = [r_{ab}(u,v),T_{a}(u)T_{b}(v)]$, how to prove that the conserved quantities are generated from $ln(\tau(u))$ in the periodic case, which are the integrable models (long-range) or condition which is not this expression which provide us the local conserved quantities in the Arnold-Liouville theorem?