Commutative-group $5$-configurations

A review of known $5$-configurations is given, for non-degenerate over the field $GF(2)$ incidence matrices $L$, like $L^{-1}$, have $5$ ones in rows and columns. A representation of them on directed graphs is given. An exhaustive description of commutative-group $5$-configurations realizable on Abelian groups is obtained, when subsets of the configuration are obtained from each other by parallel shifts.