  The problem of obtaining an explicit representation for the fourth invariant
of geodesic motion (generalized Carter constant) of an arbitrary stationary
axisymmetric vacuum spacetime generated from an Ernst Potential is considered.
The coupling between the non-local curvature content of the spacetime as
encoded in the Weyl tensor, and the existence of a Killing tensor is explored
and a constructive, algebraic test for a fourth order Killing tensor suggested.
The approach used exploits the variables defined for the B\"{a}ckland
transformations to clarify the relationship between Weyl curvature, constants
of geodesic motion, expressed as Killing tensors, and the solution generation
techniques. A new symmetric non-covariant formulation of the Killing equations
is given. This formulation transforms the problem of looking for fourth-order
Killing tensors in 4D into one of looking for four interlocking two-manifolds
admitting fourth-order Killing tensors in 2D.
