Five-dimensional electrostatic black holes in a background field

We consider $5$ dimensional electrostatic solutions to Einstein-Maxwell gravity with $2$ commuting spacelike Killing fields. Taking two distinct reductions from dimensions a $3$ base space, we write the equations using some axially symmetric functions on $\mathbb{R}^3$. These can be viewed as arising harmonic map coupled 3-dimensional isometries of target space this revealing hidden $SL(2,\mathbb{R})$ symmetry sector theory. Depending choice reduction then gives rise different 1-parameter families transformations corresponding either charging black hole or immersing it in background electric field. use these charge static Saturn and $L(n,1)$ lens spacetime by tuning strength external field, cure conical singularities give new regular solutions. Notably electrified generated is first example topologically trivial asymptotics.