It is known that any three-dimensional axisymmetric metric obeying Einstein equations can be put in the following form $ds^2=e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi $, where $\gamma=\gamma(\rho,z)$.
My question is: are there any three dimensional conformally flat metrics (besides the flat metric, corresponding to $\gamma=0$)?
I've tried to calculate Cotton tensor density explicitly, but the arising equations for $\gamma$ are of the 3rd order.