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We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3surfaces obtained by desingularization of the quotient of an abelian surface$A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$disjoint configurations of two smooth rational curves $C,C'$ with $CC'=1$. This$9{\bf A}_{2}$-configuration plays a role similar to the Nikulin configurationof $16$ disjoint smooth rational curves on (classical) Kummer surfaces. Westudy the (generalized) question of T. Shioda: suppose that Km$_{3}(A)$ isisomorphic to Km$_{3}(B)$, does that imply that $A$ and $B$ are isomorphic? Weanswer by the negative in general, by two methods: by a link between thatproblem and Fourier-Mukai partners of $A$, and by construction of $9{\bfA}_{2}$-configurations on Km$_{3}(A)$ which cannot be exchanged under theautomorphism group.

Constructions of Kummer structures on generalized Kummer surfaces