  We study multi-parameter solutions of the inhomogeneous paraxial wave
equation in a linear and quadratic approximation which include oscillating
laser beams in a parabolic waveguide, spiral light beams, and other important
families of propagation-invariant laser modes in weakly varying media. A
"smart" lens design and a similar effect of superfocusing of particle beams in
a thin monocrystal film are also discussed. In the supplementary electronic
material, we provide a computer algebra verification of the results presented
here, and of some related mathematical tools that were stated without proofs in
the literature. We also demonstrate how computer algebra can be used to derive
some of the presented formulas automatically, which is highly desirable as the
corresponding hand calculations are very tedious. In numerical simulations,
some of the new solutions reveal quite exotic properties which deserve further
investigation including an experimental observation.
