  Metric of axially symmetric asymptotically flat black holes in an arbitrary
metric theory of gravity can be represented in the general form which depends
on infinite number of parameters. We constrain this general class of metrics by
requiring the existence of additional symmetries, which lead to the separation
of variables in the Hamilton-Jacobi and Klein-Gordon equations, and show that
once the metric functions change sufficiently moderately in some region near
the black hole, the black-hole shadow depends on a few deformation parameters
only. We analyze the influence of these parameters on the black-hole shadow. We
also show that the shadow of the rotating black hole in the
Einstein-dilaton-Gauss-Bonnet theory is well approximated if the terms
violating the separation of variables are neglected in the metric.
