  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.
