  We extend the link between Einstein Sasakian manifolds and Killing spinors to
a class of $\eta$-Einstein Sasakian manifolds, both in Riemannian and
Lorentzian settings, characterising them in terms of generalised Killing
spinors. We propose a definition of supersymmetric M-theory backgrounds on such
a geometry and find a new class of such backgrounds, extending previous work of
Haupt, Lukas and Stelle.
