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We consider the families of finite Abelian groups $\ZZ/p\ZZ\times \ZZ/p\ZZ$,$\ZZ/p^2\ZZ$ and $\ZZ/p\ZZ\times \ZZ/q\ZZ$ for $p,q$ two distinct primenumbers. For the two first families we give a simple characterization of allfunctions whose support has cardinality $k$ while the size of the spectrumsatisfies a minimality condition. We do it for a large number of values of $k$in the third case. Such equality cases were previously known when $k$ dividesthe cardinality of the group, or for groups $\ZZ/p\ZZ$.

Equality cases for the uncertainty principle in finite Abelian groups