  We construct recursively an infinite number of Poisson structures for the
supersymmetric integrable hierarchy governing the Pohlmeyer reduction of
superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These
Poisson structures are all non-local and not relativistic except one, which is
the canonical Poisson structure of the semi-symmetric space sine-Gordon model
(SSSSG). We verify that the superposition of the first three Poisson structures
corresponds to the canonical Poisson structure of the reduced sigma model.
Using the recursion relations we construct commuting charges on the reduced
sigma model out of those of the SSSSG model and in the process we explain the
integrable origin of the Zukhovsky map and the twisted inner product used in
the sigma model side. Then, we compute the complete Poisson superalgebra for
the conserved Drinfeld-Sokolov supercharges associated to an exotic kind of
extended non-local rigid 2d supersymmetry recently introduced in the SSSSG
context. The superalgebra has a kink central charge which turns out to be a
generalization to the SSSSG models of the well-known central extensions of the
N=1 sine-Gordon and N=2 complex sine-Gordon model Poisson superalgebras
computed from 2d superspace. The computation is done in two different ways
concluding the proof of the existence of 2d supersymmetry in the reduced sigma
model phase space under the boost invariant SSSSG Poisson structure.
