  A new solution to the Einstein-Maxwell field equations is presented
describing a cylindrically symmetric homogeneous cosmology. The solution is
conformally flat, it possesses seven Killing vectors of which the timelike one
is rotating and one of the spacelike, pseudorotating. Our solution also admits
a Kerr-Schild form. It is alternatively produced by different electromagnetic
sources some of which represent constant null electromagnetic fields, while the
others, a circularly polarized plane electromagnetic wave (seemingly, a unique
situation in general relativity). The concrete electromagnetic four-potentials
are found from the assumption that they are proportional to the Killing
covectors. The general solution is obtained for timelike and null geodesics.
Finally, we find that this space-time admits closed timelike non-geodesic
lines.
