  We study a class of gravitational lensing systems consisting of an inclined
ring/belt, with and without an added point mass at the centre. We show that a
common feature of such systems are so-called "pseudo-caustics", across which
the magnification of a point source changes discontinuously and yet remains
finite. Such a magnification change can be associated with either a change in
image multiplicity or a sudden change in the size of a lensed image. The
existence of pseudo-caustics and the complex interplay between them and the
formal caustics (which correspond to points of infinite magnification) can lead
to interesting consequences, such as truncated or open caustics and a
non-conservation of total image parity. The origin of the pseudo-caustics is
found to be the non-differentiability of the solutions to the lens equation
across the ring/belt boundaries, with the pseudo-caustics corresponding to
ring/belt boundaries mapped into the source plane. We provide a few
illustrative examples to understand the pseudo-caustic features, and in a
separate paper consider a specific astronomical application of our results to
study microlensing by extrasolar asteroid belts.

Earth and Planetary Astrophysics;Solar and Stellar Astrophysics