  We continue to study black holes subjected to strong sources of gravity,
again paying special attention to the behaviour of geometry in the black-hole
interior. After examining, in previous two papers, the deformation arising in
the Majumdar-Papapetrou binary of extremally charged black holes and that of a
Schwarzschild black hole due to a surrounding (Bach-Weyl) ring, we consider
here the system of two Schwarzschild-type black holes held apart by the Appell
ring. After verifying that such a configuration can be in a strut-free
equilibrium along certain lines in a parameter space, we compute several basic
geometric characteristics of the equilibrium configurations. Then, like in
previous papers, we calculate and visualize simple invariants determined by the
metric (lapse or, equivalently, potential), by its first derivatives
(gravitational acceleration) and by its second derivatives (Kretschmann
scalar). Extension into the black-hole interior is achieved along particular
null geodesics starting tangentially to the horizon. In contrast to the case
involving the Bach-Weyl ring, here each single black hole is placed
asymmetrically with respect to the equatorial plane (given by the Appell ring)
and the interior geometry is really deformed in a non-symmetrical way. Inside
the black holes, we again found regions of negative Kretschmann scalar in some
cases.
