  We consider light propagation in a non-magnetized pressureless plasma around
a Kerr black hole. We find the necessary and sufficient condition the plasma
electron density has to satisfy to guarantee that the Hamilton-Jacobi equation
for the light rays is separable, i.e., that a generalized Carter constant
exists. For all cases where this condition is satisfied we determine the photon
region, i.e., the region in the spacetime where spherical light rays exist. A
spherical light ray is a light ray that stays on a sphere $r =
\mathrm{constant}$ (in Boyer-Lindquist coordinates). Based on these results, we
calculate the shadow of a Kerr black hole under the influence of a plasma that
satisfies the separability condition. More precisely, we derive an analytical
formula for the boundary curve of the shadow on the sky of an observer that is
located anywhere in the domain of outer communication. Several examples are
worked out.
