  In this paper, we study the Minkowski-type inequality for asymptotically flat
static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that
was establishedby McCormick. First, we show that any asymptotically flat static
$(M^{n},g)$ which achieves the equality and has CMC or equipotential boundary
is isometric to a rotationally symmetric region of the Schwarzschild manifold.
Then, we apply conformal techniques to derive a new Minkowski-type inequality
for the level sets of bounded static potentials. Taken together, these provide
a robust approach to detecting rotational symmetry of asymptotically flat
static systems.
  As an application, we prove global uniqueness of static metric extensions for
the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean
coordinate spheres in dimension $n < 8$ under the natural condition of
Schwarzschild stability. This generalizes an earlier result of Miao. We also
establish uniqueness for equipotential photon surfaces with small
Einstein-Hilbert energy. This is interesting to compare with other recent
uniqueness results for static photon surfaces and black holes.
