Properties of null congruences in spacetimes of Petrov type III embedded into E₅

We show that if a spacetime of Petrov type III can be embedded into E₅ then their degenerate Debever-Penrose vectors must define a null geodesic congruence which is non-rotating and non-deforming. These are necessary conditions for embedding satisfied by congruences of any class-one type-III spacetime. It is not yet known, however, whether such metrics can be generated by physically admissible energy- momentum sources or not. Using our result, we show that a metric obtained by Allnut cannot be class-one.