  We show that given a compact, connected $m$-quasi Einstein manifold $(M,g,X)$
without boundary, the potential vector field $X$ is Killing if and only if $(M,
g)$ has constant scalar curvature. This extends a result of
Bahuaud-Gunasekaran-Kunduri-Woolgar, where it is shown that $X$ is Killing if
$X$ is incompressible. We also provide a sufficient condition for a compact,
non-gradient $m$-quasi Einstein metric to admit a Killing field. We do this by
following a technique of Dunajski and Lucietti, who prove that a Killing field
always exists in this case when $m=2$. This condition provides an alternate
proof of the aforementioned result of Bahuaud-Gunasekaran-Kunduri-Woolgar. This
alternate proof works in the $m = -2$ case as well, which was not covered in
the original proof.
