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The geometries with SL$(2,\mathbb{R})$ and some axial U$(1)$ isometries arecalled ``near-horizon extremal geometries" and are found usually, but notnecessarily, in the near-horizon limit of the extremal black holes. We presenta new member of this family of solutions in five-dimensional Einstein-Hilbertgravity that has only one non-zero angular momentum. In contrast with thesingle-rotating Myers-Perry extremal black hole and its near-horizon geometryin five dimensions, this solution has a non-vanishing and finite entropy.Although there is a uniqueness theorem that prohibits the existence of suchsingle-rotating near-horizon geometries in five-dimensional general relativity,this solution has a curvature singularity at one of the poles, which breaks thesmoothness conditions in the theorem.

Single-rotating Five-dimensional Near-horizon Extremal Geometry in General Relativity