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Exact solutions in five-dimensional axi-dilaton gravity with Euler–Poincaré term

We examine the effective field equations that are obtained from the axi-dilaton gravity action with a second-order Euler–Poincaré term and a cosmological constant in all higher dimensions. We solve these equations for five-dimensional spacetimes possessing homogeneity and isotropy in their three-dimensional subspaces. For a number of interesting special cases, we show that the solutions fall into two main classes: the first class consists of time-dependent solutions with spherical or hyperboloidal symmetry which require certain fine-tuning relations between the coupling constants of the model and the cosmological constant. Solutions in the second class are locally static and prove the validity of Birkhoff's staticity theorem in the axi-dilaton gravity. We also give a class of static solutions, among them the well known charged black hole solutions with a cosmological constant in which the usual electric charge is superseded by an axion charge.