  We are concerned with the blow-up analysis of mean field equations. It has
been proven in [6] that solutions blowing-up at the same non-degenerate blow-up
set are unique. On the other hand, the authors in [18] show that solutions with
a degenerate blow-up set are in general non-unique. In this paper we first
prove that evenly symmetric solutions on a flat torus with a degenerate
two-point blow-up set are unique. In the second part of the paper we complete
the analysis by proving the existence of such blow-up solutions by using a
Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly
symmetric blow-up solutions come from one-point blow-up solutions of the mean
field equation on a "half" torus.
