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The problem of obtaining an explicit representation for the fourth invariantof geodesic motion (generalized Carter constant) of an arbitrary stationaryaxisymmetric vacuum spacetime generated from an Ernst Potential is considered.The coupling between the non-local curvature content of the spacetime asencoded in the Weyl tensor, and the existence of a Killing tensor is exploredand a constructive, algebraic test for a fourth order Killing tensor suggested.The approach used exploits the variables defined for the B\"{a}cklandtransformations to clarify the relationship between Weyl curvature, constantsof geodesic motion, expressed as Killing tensors, and the solution generationtechniques. A new symmetric non-covariant formulation of the Killing equationsis given. This formulation transforms the problem of looking for fourth-orderKilling tensors in 4D into one of looking for four interlocking two-manifoldsadmitting fourth-order Killing tensors in 2D.

Spacetime Encodings IV - The Relationship between Weyl Curvature and Killing Tensors in Stationary Axisymmetric Vacuum Spacetimes