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We construct all axi-symmetric non--gradient m--quasi--Einstein structures on a two--sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to m=2, as well as a family of new regular metrics with m 2 given in terms of hypergeometric functions. We also show that in the case m=-1 with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.
Colling et al. (Wed,) studied this question.