Abstract
Non-conformally flat spacetimes admitting a non-null two-index Killing spinor are investigated by means of the Geroch-Held-Penrose formalism. Claims appearing in the literature that such spacetimes are all explicitly known are incorrect. This was shown in [5] for the family where, in the canonical frame, the spin coefficients ρ or μ, vanish. Here the general case with non-vanishing ρ, μ, π and τ is re-considered. It is shown that the construction in [4] hinges on the tacit assumption that certain integrability conditions hold, implying two algebraic relations for the spin coefficients and the components of the Ricci spinor. All (conformal classes of) spacetimes, in which one of these conditions is violated, are obtained by invariant integration. The resulting classes are each other's Sachs transform and are characterised by one free function. They admit in general no Killing vectors, but still admit a conformal gauge (different from the trivial unitary gauge) in which a Killing tensor exists.
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