  We study the extended supersymmetric integrable hierarchy underlying the
Pohlmeyer reduction of superstring sigma models on semi-symmetric superspaces
F/G. This integrable hierarchy is constructed by coupling two copies of the
homogeneous integrable hierarchy associated to the loop Lie superalgebra
extension f of the Lie superalgebra f of F and this is done by means of the
algebraic dressing technique and a Riemann-Hilbert factorization problem. By
using the Drinfeld-Sokolov procedure we construct explicitly, a set of 2D spin
\pm1/2 conserved supercharges generating supersymmetry flows in the phase space
of the reduced model. We introduce the bi-Hamiltonian structure of the extended
homogeneous hierarchy and show that the two brackets are of the
Kostant-Kirillov type on the co-adjoint orbits defined by the light-cone Lax
operators L_\pm. By using the second symplectic structure, we show that these
supersymmetries are Hamiltonian flows, we compute part of the supercharge
algebra and find the supersymmetric field variations they induce. We also show
that this second Poisson structure coincides with the canonical
Lorentz-Invariant symplectic structure of the WZNW model involved in the
Lagrangian formulation of the extended integrable hierarchy, namely, the
semi-symmetric space sine-Gordon model (SSSSG), which is the Pohlmeyer reduced
action functional for the transverse degrees of freedom of superstring sigma
models on the cosets F/G. We work out in some detail the Pohlmeyer reduction of
the AdS_2xS^2 and the AdS_3xS^3 superstrings and show that the new conserved
supercharges can be related to the supercharges extracted from 2D superspace.
In particular, for the AdS_2xS^2 example, they are formally the same.
