On some moments of planar convex bodies - 2

Let us begin with a question that is under discussion. Thanks to Professors Roman Karasev and Arseniy Akopyan

1. An ellipse is usually defined through the pair of its foci lying on its major axis. Is there a natural way to define an ellipse through a pair of points lying on its minor axis?
Note: We mean real points. In particular, there must be two points on the minor axis of the ellipse such that the inertia form (the second moment) of the solid ellipse about those points is round (is a scalar matrix).
Of course, it may well happen that this analytic definition does not correspond to anything geometric. And let me add, this question got deleted from mathoverflow for reasons I don't understand.
---------------
2. Let us define a moment bisector to be a line l thru a planar region C that cuts it into two pieces with equal moment (of inertia or defined in any way given in last post here) about l itself.
Question: What can we say about points on C that have more than one moment bisector thru it?