Problem. Several circles are drawn on the plane and all points of their intersection or touching are marked. For which $n$ it is possible that each circle contains exactly $n$ marked points and each point belongs to exactly $n$ circles?

Examples for $n=2,3$ are trivial. For $n=4$ and $5$ there only two known examples on the figures below (we apply an inversion with a center not lying on these lines to obtain the required configurations). Are there examples for $n>5$ and other examples for $n=4,5$?