  This article concerns the number of nodal domains of eigenfunctions of the
Laplacian on special Riemannian $3$-manifolds, namely nontrivial principal
$S^1$ bundles $P \to X$ over Riemann surfaces equipped with certain $S^1$
invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein
metrics that any Laplacian eigenfunction has exactly two nodal domains unless
it is invariant under the $S^1$ action.
  We also construct an explicit orthonormal eigenbasis on the flat $3$-torus
$\mathbb{T}^3$ for which every non-constant eigenfunction belonging to the
basis has two nodal domains.
