  Very-long baseline interferometric observations have resolved structure on
scales of only a few Schwarzschild radii around the supermassive black holes at
the centers of our Galaxy and M87. In the near future, such observations are
expected to image the shadows of these black holes together with a bright and
narrow ring surrounding their shadows. For a Kerr black hole, the shape of this
photon ring is nearly circular unless the black hole spins very rapidly.
Whether or not, however, astrophysical black holes are truly described by the
Kerr metric as encapsulated in the no-hair theorem still remains an untested
assumption. For black holes that differ from Kerr black holes, photon rings
have been shown numerically to be asymmetric for small to intermediate spins.
In this paper, I calculate semi-analytic expressions of the shapes of photon
rings around black holes described by a new Kerr-like metric which is valid for
all spins. I show that photon rings in this spacetime are affected by two types
of deviations from the Kerr metric which can cause the ring shape to be highly
asymmetric. I argue that the ring asymmetry is a direct measure of a potential
violation of the no-hair theorem and that both types of deviations can be
detected independently if the mass and distance of the black hole are known. In
addition, I obtain approximate expressions of the diameters, displacements, and
asymmetries of photon rings around Kerr and Kerr-like black holes.
