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 $m\neq 2$ 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.