  We present different proofs of the uniqueness of 4-dimensional static vacuum
asymptotically flat spacetimes containing a connected equipotential photon
surface or in particular a connected photon sphere. We do not assume that the
equipotential photon surface is outward directed or non-degenerate and hence
cover not only the positive but also the negative and the zero mass case which
has not yet been treated in the literature. Our results partially reproduce and
extend beyond results by Cederbaum and by Cederbaum and Galloway. In the
positive and negative mass cases, we give three proofs which are based on the
approaches to proving black hole uniqueness by Israel, Robinson, and
Agostiniani--Mazzieri, respectively. In the zero mass case, we give four
proofs. One is based on the positive mass theorem, the second one is inspired
by Israel's approach and in particular leads to a new proof of the Willmore
inequality in $(\mathbb{R}^3, \delta)$, under a technical assumption. The
remaining two proofs are inspired by proofs of the Willmore inequality by
Cederbaum and Miehe and by Agostiniani and Mazzieri, respectively. In
particular, this suggests to view the Willmore inequality and its rigidity case
as a zero mass version of equipotential photon surface uniqueness.
