  Several new mutation-periodic quivers of period higher than 1 are introduced
as well as the associated discrete dynamical systems. The reduction of these
systems is developed using either a presymplectic or a Poisson approach. The
presymplectic approach leads to a reduced system whose iteration map is
symplectic with respect to a log symplectic form. In the Poisson approach we
build a Poisson structure invariant under the iteration map, leading to a
reduced system whose variables are the Casimirs of such structure.
