  The motion of photons around black holes determines the shape of shadow and
match the ringdown properties of a perturbed black hole. Observations of
shadows and ringdown waveforms will reveal the nature of black holes. In this
paper, we study the motion of photons in a general parametrized metric beyond
the Kerr hypothesis. We investigated the radius and frequency of the photon
circular orbits on the equatorial plane and obtained fitted formula with varied
parameters. The Lyapunov exponent which connects to the decay rate of the
ringdown amplitude is also calculated. We also analyzed the shape of shadow
with full parameters of the generally axisymmetric metric. Our results imply
the potential constraint on black hole parameters by combining the Event
Horizon Telescope and gravitational wave observations in the future.
