Third-order geometrical null symmetries of gravitational fields

The first-, the second-, and the third-order null Killing vectors in a gravitational field are explored separately. When an algebraically special Petrov-type free gravitational field isaligned with a source-free (nonnull/null) electromagnetic field, their common propagation vectorl^a is shown to be a third-order null Killing vector field (_lllg_ab = 0,_llg_ab 0). When the two fields arenot aligned, the necessary and sufficient conditions for the existence of a third-order null symmetry are obtained in Newman-Penrose formalism.