Can a stationary Bianchi black brane have momentum along the direction with no translational symmetry?

Bianchi black branes (black brane solutions with homogeneous but anisotropic horizons classified by the Bianchi type) provide a simple holographic setting with lattice structures taken into account. In the case of holographic superconductor, we have a persistent current with lattices. Accordingly, we expect that in the dual gravity side, a black brane should carry some momentum along a direction of lattice structure, where translational invariance is broken. Motivated by this expectation, we consider whether - and if possible, in what circumstances - a Bianchi black brane can have momentum along a direction of no-translational invariance. First, we show that this cannot be the case for a certain class of stationary Bianchi black brane solutions in the Einstein-Maxwell-dilation theory. Then we also show that this can be the case for some Bianchi VII₀ black branes by numerically constructing such a solution in the Einstein-Maxwell theory with an additional vector field having a source term. The horizon of this solution admits a translational invariance on the horizon and conveys momentum (and is "rotating" when compactified). However this translational invariance is broken just outside the horizon. This indicates the existence of a black brane solution which is regular but non-analytic at the horizon, thereby evading the black hole rigidity theorem.