  This article is concerned with the question of uniqueness of self-similar
profiles for Smoluchowski's coagulation equation which exhibit algebraic decay
(fat tails) at infinity. More precisely, we consider a rate kernel $K$ which
can be written as $K=2+\varepsilon W$. The perturbation is assumed to have
homogeneity zero and might also be singular both at zero and at infinity. Under
further regularity assumptions on $W$, we will show that for sufficiently small
$\varepsilon$ there exists, up to normalisation of the tail behaviour at
infinity, at most one self-similar profile.
