  The general line element corresponding to the family of algebraically
general, gravito-electric, expanding, rotating dust models with one
functionally independent zero-order Riemann invariant is constructed. The
isometry group is at most one-dimensional but generically trivial. It is shown
that the asymmetric solutions with constant ratio of energy density and
vorticity amplitude provide first examples of Petrov type I space-times for
which the Karlhede classification requires the computation of the third
covariant derivative of the Riemann tensor.
