  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 $VI_0$, has degenerate shear and is of Petrov type
I($M^\infty)$ 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.
