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Recently the class of purely magnetic non-rotating dust spacetimes has been shown to be empty (Wylleman, Class. Quant. Grav. 23, 2727). It turns out that purely magnetic rotating dust models are subject to severe integrability conditions as well. One of the consequences of the present paper is that also rotating dust cannot be purely magnetic when it is of Petrov type D or when it has a vanishing spatial gradient of the energy density. For purely magnetic and non-rotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type D (Lozanovski, Class. Quant. Grav. 19, 6377), the fluid is shown to be non-accelerating if and only if the spatial density gradient vanishes. Under these conditions, a new and algebraically general solution is found, which is unique up to a constant rescaling, which is spatially homogeneous of Bianchi type V I0, has degenerate shear and is of Petrov type I(M ∞ ) in the extended Arianrhod-McIntosh classification.The metric and the equation of state are explicitly constructed and properties of the model are briefly discussed. We finally situate it within the class of normal geodesic flows with degenerate shear tensor.

A classification of Petrov-type D Killing spinor spacetimes admitting a homogeneous conformal representant is presented. For each class a canonical line element is constructed and a physical interpretation of its conformal members is discussed.

Spacetimes admitting non-null valence two Killing spinors are discussed. An explicit example of such a spacetime is given which shows that, contrary to claims appearing in the literature, not all of these spacetimes have been listed exhaustively. We also give conditions for determining whether a given spacetime is conformally related to a Killing-Yano spacetime.