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We consider Einstein gravity coupled to an n-form field strength in Ddimensions. Such a theory cannot be supersymmetrized in general, wenevertheless propose a pseudo-Killing spinor equation and show that the AdS XSphere vacua have the maximum number of pseudo-Killing spinors, and hence arefully pseudo-supersymmetric. We show that extremal p-branes and theirintersecting configurations preserve fractions of the pseudo-supersymmetry. Westudy the integrability condition for general (D,n) and obtain the additionalconstraints that are required so that the existence of the pseudo-Killingspinors implies the Einstein equations of motion. We obtain newpseudo-supersymmetric bubbling AdS_5 X S^5 spaces that are supported by anon-self-dual 5-form. This demonstrates that non-supersymmegtric conformalfield theories may also have bubbling states of arbitrary droplets of freefermions in the phase space. We also obtain an example of less-bubbling AdSgeometry in D=8, whose bubbling effects are severely restricted by theadditional constraint arising from the integrability condition.

Pseudo-Killing Spinors, Pseudo-supersymmetric p-branes, Bubbling and Less-bubbling AdS Spaces