Backlund transformations and Hamiltonian flows

In this work we show that, under certain conditions, parametric Backlund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation lead to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, that indeed will be a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An application to the Toda periodic lattice is given.