  Conformal invariance of two-dimensional variational problems is a condition
known to enable a blow-up analysis of solutions and to deduce the removability
of singularities. In this paper, we identify another condition that is not only
sufficient, but also necessary for such a removability of singularities. This
is the validity of the Pohozaev identity. In situations where such an identity
fails to hold, we introduce a new quantity, called the {\it Pohozaev constant},
which on one hand measures the extent to which the Pohozaev identity fails and
on the other hand provides a characterization of the singular behavior of a
solution at an isolated singularity. We apply this to the blow-up analysis for
super-Liouville type equations on Riemann surfaces with conical singularities,
because in the presence of such singularities, conformal invariance no longer
holds and a local singularity is in general non-removable unless the Pohozaev
constant is vanishing.
