  A Nikulin configuration is the data of $16$ disjoint smooth rational curves
on a K3 surface. According to results of Nikulin, the existence of a Nikulin
configuration means that the K3 surface is a Kummer surface, moreover the
abelian surface from the Kummer structure is determined by the $16$ curves. A
classical question of Shioda is about the existence of non isomorphic Kummer
structures on the same Kummer K3 surface. The question was studied by several
authors, and it was shown that the number of non-isomorphic Kummer structures
is finite, but no explicit geometric construction of such structures was given.
In a previous paper, we constructed explicitly non isomorphic Kummer structures
on some Kummer surfaces. In this paper we generalise the construction to Kummer
surfaces with a weaker restriction on the degree of the polarization and we
describe some cases where the previous construction does not work.
