  This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV)
spacetimes that were found by Carter to have separable Hamilton-Jacobi
equations, and thus admit a second-order Killing tensor. The derivation of the
spacetimes presented in this paper borrows from ideas about dynamical systems,
and illustrates concepts that can be generalized to higher- order Killing
tensors. The relationship between the components of the Killing equations and
metric functions are given explicitly. The origin of the four separable
coordinate systems found by Carter is explained and classified in terms of the
analytic structure associated with the Killing equations. A geometric picture
of what the orbital invariants may represent is built. Requiring that a SAV
spacetime admits a second-order Killing tensor is very restrictive, selecting
very few candidates from the group of all possible SAV spacetimes. This
restriction arises due to the fact that the consistency conditions associated
with the Killing equations require that the field variables obey a second-order
differential equation, as opposed to a fourth-order differential equation that
imposes the weaker condition that the spacetime be SAV. This paper introduces
ideas that could lead to the explicit computation of more general orbital
invariants in the form of higher-order Killing Tensors.
