  A long standing problem is the supergauge completion of AdS_4 x (G/H)_7 or
AdS_5 x (G/H)_5 backgrounds which preserve less then maximal supersymmetry. In
parallel with the supersolvable realization of the AdS_4 x S^7 background based
on Kappa-symmetry, we develop a technique which amounts to solving the
above-mentioned problem in a way useful for pure spinor quantization for
supermembranes and superstrings. Instead of gauge fixing some of the superspace
coordinates to zero, we impose an additional constraint on them reproducing the
simplifications of the supersolvable representations. The constraints are
quadratic, homogeneous, Sp(4,R)-covariant, and consistent from the quantum
point of view in the pure spinor approach. Here we provide the geometrical
solution which, in a subsequent work, will be applied to the membrane and the
superstring sigma models.
