  We prove a KAM-type result for the persistence of two-dimensional invariant
tori in perturbations of integrable action-angle-angle maps with degeneracy,
satisfying the intersection property.
  Such degenerate action-angle-angle maps arise upon generic perturbation of
three-dimensional volume-preserving vector fields, which are invariant under
volume-preserving action of $S^1$ when there is no motion in the group action
direction for the unperturbed map. This situation is analogous to degeneracy in
Hamiltonian systems. The degenerate nature of the map and the unequal number of
action and angle variables make the persistence proof non-standard. The
persistence of the invariant tori as predicted by our result has implications
for the existence of barriers to transport in three-dimensional incompressible
fluid flows. Simulation results indicating existence of two-dimensional tori in
a perturbation of swirling Hill's spherical vortex flow are presented.
