New vacuum solutions of conformal Weyl gravity

The Bach equation, i.e., the vacuum field equation following from the Lagrangian $\mathrm{L}{\mathrm{=C}}_{\mathrm{ijkl}}{C}^{\mathrm{ijkl}},$ will be completely solved for the case that the metric is conformally related to the Cartesian product of two two-spaces; this covers the spherically and the plane symmetric space–times as special subcases. Contrary to other approaches, we make a covariant $\text{2+2}$ -decomposition of the field equation, and so we are able to apply results from two-dimensional gravity. Finally, some cosmological solutions will be presented and discussed.