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An invariant classification of algebraically general rotating dust models with a purely electric Weyl tensor is presented. These models are thus 'Newtonian' and '1+3-covariantly silent', in the senses described by Ellis, Maartens, van Elst et al. It turns out that the vorticity vector is necessarily parallel to a geodesic spatial Weyl eigenvector.

Metric expressions in invariantly built coordinates are given for certain subcases. Using the Janet-Riquier theory for partial differential equations we discuss the number of free functions in the (not explicitly constructible) general solution. In this respect, a fundamental distinction between the expanding and non-expanding case naturally arises.