  Recently, the authors have formulated and explored a novel
Painleve-Gullstrand variant of the Lense-Thirring spacetime, which has some
particularly elegant features -- including unit-lapse, intrinsically flat
spatial 3-slices, and some particularly simple geodesics, the "rain" geodesics.
At linear level in the rotation parameter this spacetime is indistinguishable
from the usual slow-rotation expansion of Kerr. Herein, we shall show that this
spacetime possesses a nontrivial Killing tensor, implying separability of the
Hamilton-Jacobi equation. Furthermore, we shall show that the Klein-Gordon
equation is also separable on this spacetime. However, while the Killing tensor
has a 2-form square root, we shall see that this 2-form square root of the
Killing tensor is not a Killing-Yano tensor. Finally, the
Killing-tensor-induced Carter constant is easily extracted, and now, with a
fourth constant of motion, the geodesics become (in principle) explicitly
integrable.
