Homogeneous Anisotropic Black Branes with Bianchi VI$_h$ Symmetry

We construct a new family of exact vacuum black brane solutions to five-dimensional Einstein gravity with a negative cosmological constant, characterized by a homogeneous horizon with Bianchi VI$_h$ symmetry. This construction generalizes the known Solv (Bianchi VI$_{-1}$) geometry via a continuous anisotropy parameter, $h$. By reducing the field equations to a cohomogeneity-one system, we derive the metric analytically. These homogeneous solutions are not asymptotically (locally) AdS, but nevertheless we analyze the thermodynamics, establishing scaling relations between entropy and temperature with anisotropic holographic system in mind. Additionally, we identify a new branch of Ricci-flat, hyperscaling-violating vacuum solutions in the case where the cosmological constant vanishes.