Selecting Horndeski theories without apparent symmetries and their black hole solutions

Starting from a generalised Kaluza-Klein action including arbitrary Horndeski potentials, we establish integrability and compatibility conditions that solve the generic field equations for spherical symmetry. The resulting theories can be identified as general Horndeski theories having no apparent symmetries in four dimensions or as effective string theory actions with an IR logarithmic running for the dilaton, higher order corrections and generalised Liouville type potentials. For such actions, we then find black holes with secondary hair parameterized by two coupling constants essentially characterising the theories at hand. One is related to an action which is conformally coupled in five dimensions while the second is related to a Kaluza-Klein reduction of Lovelock theory. We show that the full action can also be interpreted as a sum of conformally coupled actions in differing dimensions. Known solutions are mapped within the general chart of the found theories and novel general black holes are discussed, focusing on their important properties and some of their observational constraints.