  Herein we explore the non-equatorial constant-$r$ ("quasi-circular")
geodesics (both timelike and null) in the Painleve-Gullstrand variant of the
Lense-Thirring spacetime recently introduced by the current authors. Even
though the spacetime is not spherically symmetric, shells of constant-$r$
geodesics still exist. Whereas the radial motion is (by construction) utterly
trivial, determining the allowed locations of these constant-$r$ geodesics is
decidedly non-trivial, and the stability analysis is equally tricky. Regarding
the angular motion, these constant-$r$ orbits will be seen to exhibit both
precession and nutation -- typically with incommensurate frequencies. Thus this
constant-$r$ geodesic motion, though integrable in the precise technical sense,
is generically surface-filling, with the orbits completely covering a symmetric
equatorial band which is a segment of a spherical surface, (a so-called
"spherical zone"), and whose latitudinal extent is governed by delicate
interplay between the orbital angular momentum and the Carter constant. The
situation is qualitatively similar to that for the (exact) Kerr spacetime --
but we now see that any physical model having the same slow-rotation weak-field
limit as general relativity will still possess non-equatorial constant-$r$
geodesics.
