Pick two points $(x,0)$ and $(0,y)$ (say $x>0$ and $y>0$). Pick a unit vector $u = (u_1,u_2)$, $v = (v_1, v_2)$, and attach one to each of the points. Provided $u$ and $v$ are "nice" ($v$ needs to lie "between" $(u_1,u_2)$ and $(-u_1,u_2)$ in the appropriate sense -- in their convex hull if $u_2 > 0$, outside of it if $u_2 < 0$, and $v_1 > 0$ if $u_2 = 0$), there exists a unique ellipse with major axes parallel to the coordinate axes, passing through the two points, and with the unit vectors as its unit outer normals. (I think.)
Is there a generalization of this statement to higher dimensions?