We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus in contrast to the case of parallel spinors the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.
Craig van COEVERING. "Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds." J. Math. Soc. Japan 69 (1) 53 - 91, January, 2017. https://doi.org/10.2969/jmsj/06910053