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Stationary and axially symmetric space-times play an important role inastrophysics, particularly in the theory of neutron stars and black holes. Thestatic vacuum sub-class of these space-times is known as Weyl's class, andcontains the Schwarzschild space-time as its most prominent example. This paperis going to study the space of Killing tensor fields of valence 3 forspace-times of Weyl's class. Killing tensor fields play a crucial role inphysics 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 axiallysymmetric vacuum space-times the space of Killing tensor fields of valence 3 isgenerated by Killing vector fields and quadratic Killing tensor fields. Usingthis result, it will be proven that for the family of Zipoy-Voorhees metrics,valence-3 Killing tensor fields are always generated by Killing vector fieldsand the metric.

Reducibility of Valence-3 Killing Tensors in Weyl's Class of Stationary and Axially Symmetric Space-Times