  We consider Einstein gravity coupled to an n-form field strength in D
dimensions. Such a theory cannot be supersymmetrized in general, we
nevertheless propose a pseudo-Killing spinor equation and show that the AdS X
Sphere vacua have the maximum number of pseudo-Killing spinors, and hence are
fully pseudo-supersymmetric. We show that extremal p-branes and their
intersecting configurations preserve fractions of the pseudo-supersymmetry. We
study the integrability condition for general (D,n) and obtain the additional
constraints that are required so that the existence of the pseudo-Killing
spinors implies the Einstein equations of motion. We obtain new
pseudo-supersymmetric bubbling AdS_5 X S^5 spaces that are supported by a
non-self-dual 5-form. This demonstrates that non-supersymmegtric conformal
field theories may also have bubbling states of arbitrary droplets of free
fermions in the phase space. We also obtain an example of less-bubbling AdS
geometry in D=8, whose bubbling effects are severely restricted by the
additional constraint arising from the integrability condition.
