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We prove the existence of a family of blow-up solutions of a mean field equation on the sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green’s function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetryTd{T_{d}}which is isomorphic to the symmetric groupS4{S_{4}}. Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons and dodecahedrons). All of these solutions have the symmetries of the corresponding configuration, while they are non-axially symmetric.

Non-axially symmetric solutions of a mean field equation on 𝕊 2