Stationary and axially symmetric spacetimes play an important role in astrophysics, particularly in the theory of neutron stars and black holes. The static vacuum subclass of these spacetimes is known as Weyl’s class, and contains the Schwarzschild spacetime as its most prominent example. This paper is going to study the space of Killing tensor fields of valence 3 for spacetimes of Weyl’s class. Killing tensor fields play a crucial role in physics since they are in correspondence to invariants of the geodesic motion (i.e. constants of the motion). It will be proven that in static and axially symmetric vacuum spacetimes the space of Killing tensor fields of valence 3 is generated by Killing vector fields and quadratic Killing tensor fields. Using this result, it will be proven that for the family of Zipoy-Voorhees metrics, valence-3 Killing tensor fields are always generated by Killing vector fields and the metric.

• Received 25 June 2015

DOI:https://doi.org/10.1103/PhysRevD.92.084036