  As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed
manifolds and near-horizon geometries of extreme black holes, we study such
spaces when the vector field $X$ is divergence-free but not identically zero.
This condition is satisfied by left-invariant quasi-Einstein metrics on compact
homogeneous spaces (including the near-horizon geometry of an extreme
Myers-Perry black hole with equal angular momenta in two distinct planes), and
on certain bundles over K\"ahler-Einstein manifolds. We find that these spaces
exhibit a mild form of rigidity: they always admit a one-parameter group of
isometries generated by $X$. Further geometrical and topological restrictions
are also obtained.
