  In this paper we continue the study, initiated in arXiv:1009.3498 and
arXiv:1104.1793, of the classical integrability of Green-Schwarz superstrings
in AdS(4) x CP(3) and AdS(2) x S(2) x T(6) superbackgrounds whose spectrum
contains non-supercoset worldsheet degrees of freedom corresponding to broken
supersymmetries in the bulk. We derive an explicit expression, to all orders in
the coset fermions and to second order in the non-coset fermions, which extends
the supercoset Lax connection in these backgrounds with terms depending on the
non-coset fermions. An important property of the obtained form of the Lax
connection is that it is invariant under Z_4-transformations of the
superisometry generators and the spectral parameter. This demonstrates that the
contribution of the non-coset fermions does not spoil the Z_4-symmetry of the
super-coset Lax connection which is of crucial importance for the application
of Bethe-ansatz techniques. The expressions describing the AdS(4) x CP(3) and
AdS(2) x S(2) x T(6) superstring sigma--models and their Lax connections have a
very similar form. This is because their amount of target-space supersymmetries
complement each other to 32=24+8, the maximal number of 10d type II
supersymmetries. As a byproduct, this similarity has allowed us to obtain the
form of the geometry of the complete type IIA AdS(2) x S(2) x T(6) superspace
to all orders in the coset fermions and to the second order in the non-coset
ones.
