  We investigate an N-state spin model called quantum relativistic Toda chain
and based on the unitary finite dimensional representations of the Weyl algebra
with q being N-th primitive root of unity. Parameters of the finite dimensional
representation of the local Weyl algebra form the classical discrete integrable
system. Nontrivial dynamics of the classical counterpart corresponds to
isospectral transformations of the spin system. Similarity operators are
constructed with the help of modified Baxter's Q-operators. The classical
counterpart of the modified Q-operator for the initial homogeneous spin chain
is a Baecklund transformation. This transformation creates an extra Hirota-type
soliton in a parameterization of the chain structure. Special choice of values
of solitonic amplitudes yields a degeneration of spin eigenstates, leading to
the quantum separation of variables, or the functional Bethe ansatz. A
projector to the separated eigenstates is constructed explicitly as a product
of modified Q-operators.
