  The aim of this note is to present the first qualitative global bifurcation
diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$. To this end, we
introduce the notion of domains of first/second kind for singular mean field
equations and base our approach on a suitable spectral analysis. In particular,
we treat also non-radial solutions and non-symmetric domains and show that the
shape of the branch of solutions still resembles the well-known one of the
model regular radial case on the disk. Some work is devoted also to the
asymptotic profile for $\mu\to-\infty$.
