  Using the n-particle periodic Toda lattice and the relativistic
generalization due to Ruijsenaars of the elliptic Calogero-Moser system as
examples, we revise the basic properties of the Baecklund transformations
(BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's
quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a
conjugated variable mu for the parameter lambda of the BT B_lambda such that mu
belongs to the spectrum of the Lax operator L(lambda). As a consequence, the
generating function of the composition of n BT's gives rise also to another
canonical transformation separating variables for the model. For the Toda
lattice the dual BT parametrized by mu is introduced.

Exactly Solvable and Integrable Systems;Exactly Solvable and Integrable Systems