  We prove that the real parts of equivariant (but non-invariant)
eigenfunctions of generic bundle metrics on principal $S^1$ bundles over
manifolds of any dimension have connected nodal sets and exactly 2 nodal
domains. This generalizes earlier results of the authors in the 3-dimensional
case. The failure of the results on for non-free $S^1$ actions is illustrated
on even dimensional spheres by one-parameter subgroups of rotations whose fixed
point set consists of two antipodal points.

Spectral Theory;Analysis of PDEs