  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.
