  We generalize our recent method for constructing Killing tensors of the
second rank to conformal Killing tensors. The method is intended for foliated
spacetimes of arbitrary dimension $m$, which have a set of conformal Killing
vectors. It applies to foliations of a more general structure than in previous
literature. The basic idea is to start with reducible Killing tensors in slices
constructed from a set of conformal Killing vectors and the induced metric, and
then lift them to the whole manifold. Integrability conditions are derived that
ensure this, and a constructive lifting procedure is presented. The resulting
conformal Killing tensor may be irreducible. It is shown that subdomains of
foliation slices suitable for the method are fundamental photon surfaces if
some additional photon region inequality is satisfied. Thus our procedure also
opens the way to obtain a simple general analytical expression for the boundary
of the gravitational shadow. We apply this technique to electrovacuum, and
${\cal N}=2,\,4,\,8$ supergravity black holes, providing a new easy way to
establish the existence of exact and conformal Killing tensors.
