Symmetries and the Karlhede classification of type \(N\) vacuum solutions (Q2704074)

In solving the equivalence problem for the spacetime metrics one must calculate successive covariant derivatives of the Riemann tensor up to the upper bound order. Using the GHP formalism (Geroch Held Penrose) in the Karlhede algorithm to express the Weyl spinor and its derivatives one of the authors and Vickers reduced the bound of differentiation for Petrov type \(N\) empty spacetimes to five. In this paper the authors show that the existence of two independent symmetries (Killing vector fields/homothetic vector fields) reduces the upper bound on the Karlhede algorithm from five to three, as long as the vector fields obey a special geometric condition. The obtained results are checked in the case of the Hauser metric.