Canonicity of Bäcklund Transformation: $r$-Matrix Approach. II

This work represents the second part of the paper devoted to the general proof of the canonicity of the Bäcklund transformation (BT) for Hamiltonian integrable systems described by an $SL(2)$-invariant $r$-matrix. Introducing an extended phase space from which the original space is obtained by imposing first-kind constraints, one can prove the canonicity of the BT by a new method. This new proof provides a natural explanation for the fact why the gauge transformation of the matrix $M$ associated with the BT has the same structure as the Lax operator $L$. This technique is illustrated through an example of a DST chain.