A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration C, then X is a Kummer surface X = Km(B) where B is an Abelian surface determined by C. Let B be a generic Abelian surface having a polarization M with M 2 = k(k + 1) (for k > 0 an integer) and let X = Km(B) be the associated Kummer surface. To the natural Nikulin configuration C on X = Km(B), we associate another Nikulin configuration C ; we denote by B the Abelian surface associated to C , so that we have also X = Km(B). For k ≥ 2 we prove that B and B are not isomorphic. We then construct an infinite order automorphism of the Kummer surface X that occurs naturally from our situation. Associated to the two Nikulin configurations C, C , there exists a natural bi-double cover S → X, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for k = 2 is a Schoen surface.