  We study the integrability of a family of birational maps obtained as
reductions of the discrete Hirota equation, which are related to travelling
wave solutions of the lattice KdV equation. In particular, for reductions
corresponding to waves moving with rational speed N/M on the lattice, where N,M
are coprime integers, we prove the Liouville integrability of the maps when N +
M is odd, and prove various properties of the general case. There are two main
ingredients to our construction: the cluster algebra associated with each of
the Hirota bilinear equations, which provides invariant (pre)symplectic and
Poisson structures; and the connection of the monodromy matrices of the
dressing chain with those of the KdV travelling wave reductions.
