  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 $\mathcal{C}$, then $X$ is a Kummer surface
$X=Km(B)$ where $B$ is an Abelian surface determined by $\mathcal{C}$. 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 $\mathcal{C}$ on $X=Km(B)$, we associate another
Nikulin configuration $\mathcal{C}'$; we denote by $B'$ the Abelian surface
associated to $\mathcal{C}'$, so that we have also $X=Km(B')$. For $k\geq2$ 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 $\mathcal{C},$
$\mathcal{C}'$, there exists a natural bi-double cover $S\to X$, 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.
