  We investigate properties of higher-dimensional generally rotating black-hole
spacetimes, so called Kerr-NUT-(A)dS spacetimes, as well as a family of related
spaces which share the same explicit and hidden symmetries. In these spaces, we
study a particle motion in the presence of a weak electromagnetic field and
compare it with its operator analogies. First, we find general commutativity
conditions for classical observables and for their operator counterparts, then
we investigate a fulfillment of these conditions in the Kerr-NUT-(A)dS and
related spaces. We find the most general form of the weak electromagnetic field
compatible with the complete integrability of the particle motion and the
comutativity of the field operators. For such a field we solve the charged
Hamilton-Jacobi and Klein-Gordon equations by separation of variables.
