  Let a torus $T$ act freely on a closed manifold $M$ of dimension at least
two. We demonstrate that, for a generic $T$-invariant Riemannian metric $g$ on
$M$, each real $\Delta_g$-eigenspace is an irreducible real representation of
$T$ and, therefore, has dimension at most two. We also show that, for the
generic $T$-invariant metric on $M$, if $u$ is a non-invariant real-valued
$\Delta_g$-eigenfunction that vanishes on some $T$-orbit, then the nodal set of
$u$ is a connected smooth hypersurface whose complement has exactly two
connected components.
