On the integrability of Einstein-Maxwell-(A)dS gravity in presence of Killing vectors

We study some symmetry and integrability properties of four-dimensional Einstein-Maxwell gravity with nonvanishing cosmological constant in the presence of Killing vectors. First of all, we consider stationary spacetimes, which lead, after a timelike Kaluza-Klein reduction followed by a dualization of the two vector fields, to a three-dimensional nonlinear sigma model coupled to gravity, whose target space is a noncompact version of $\mathbb{C}\text{P}^2$ with SU(2,1) isometry group. It is shown that the potential for the scalars, that arises from the cosmological constant in four dimensions, breaks three of the eight SU(2,1) symmetries, corresponding to the generalized Ehlers and the two Harrison transformations. This leaves a semidirect product of a one-dimensional Heisenberg group and a translation group $\mathbb{R}^2$ as residual symmetry. We show that, under the additional assumptions that the three-dimensional manifold is conformal to a product space $\mathbb{R}\times\Sigma$, and all fields depend only on the coordinate along $\mathbb{R}$, the equations of motion are integrable. This generalizes the results of Leigh et al. in arXiv:1403.6511 to the case where also electromagnetic fields are present. In the second part of the paper we consider the purely gravitational spacetime admitting a second Killing vector that commutes with the timelike one. We write down the resulting two-dimensional action and discuss its symmetries. If the fields depend only on one of the two coordinates, the equations of motion are again integrable, and the solution turns out to be one constructed by Krasinski many years ago.