Here's an example of an order $96_6$ configuration found by L.W. Berman. Every point has six lines, every line has six points.
The unique 6,6 cage graph is bipartite and is a Levi graph for the following set of 31 lines.
{{1,2,4,9,13,19},{1,3,8,12,18,31},{1,5,11,24,25,27},{1,6,10,16,29,30},{1,7,20,21,23,28},{1,14,15,17,22,26},{2,3,5,10,14,20},{2,6,12,25,26,28},{2,7,11,17,30,31},{2,8,21,22,24,29},{2,15,16,18,23,27},{3,4,6,11,15,21},{3,7,13,26,27,29},{3,9,22,23,25,30},{3,16,17,19,24,28},{4,5,7,12,16,22},{4,8,14,27,28,30},{4,10,23,24,26,31},{4,17,18,20,25,29},{5,6,8,13,17,23},{5,9,15,28,29,31},{5,18,19,21,26,30},{6,7,9,14,18,24},{6,19,20,22,27,31},{7,8,10,15,19,25},{8,9,11,16,20,26},{9,10,12,17,21,27},{10,11,13,18,22,28},{11,12,14,19,23,29},{12,13,15,20,24,30},{13,14,16,21,25,31}}
Is there some way of arranging 31 points so that the corresponding pseudolines/splines/curves/lines/ellipses going the points is reasonably nice and symmetric?
EDIT: It occurs to me that these lines might be equivalent to the order-31 difference set generated by {1, 5, 11, 24, 25, 27}. That turns out to be true, here's the 6,6 cage with a difference-set based embedding.
