  We present a new method for generating the gravitational field of thin disks
within the Weyl class of static and axially symmetric spacetimes. Such a
gravitational field is described by two metric functions: one satisfies the
Laplace equation and represents the gravitational potential, while the other is
determined by line integration. We show how to obtain analytic thin-disk
solutions by convolving a certain weight function -- an Abel transformation of
the physical surface-density profile -- with the Appell-ring potential. We thus
re-derive several known thin-disk solutions while, in some cases, completing
the metric by explicitly computing the second metric function. Additionally, we
obtain the total gravitational field of several superpositions of a disk with
the Schwarzschild black hole. While the superposition problem is simple
(linear) for the potential, it is mostly not such for the second metric
function. However, in particular cases, both metric functions of the
superposition can be found explicitly. Finally, we discuss a simpler procedure
which yields the potentials of power-law-density disks we studied recently.
