  A new method for separating variables in Maxwell's equations in four- and
higher-dimensional Kerr-(A)dS spacetimes proposed recently by Lunin is
generalized to any off-shell metric that admits a principal Killing-Yano
tensor. The key observation is that Lunin's ansatz for the vector potential can
be formulated in a covariant form - in terms of the principal tensor. In
particular, focusing on the four-dimensional case we demonstrate separability
of Maxwell's equations in the Kerr-NUT-(A)dS and the Plebanski-Demianski family
of spacetimes. The new method of separation of variables is quite different
from the standard approach based on the Newman-Penrose formalism.
