What is the Orientation of Lightcones in a Kerr Ring?

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I am trying to find any diagrams like the following ones:

http://www.valdostamuseum.org/hamsmith/DFblackIn.gif
http://www.illc.uva.nl/~seop/entries/sp ... one-bh.gif
http://www.orbiter-forum.com/gallery/da ... ration.jpg

(the 1st is the best, even I failed to find it in better resolution)

But for the Kerr ring?
Especially, how these lightcones are oriented inside the ring?
 
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Could anyone at least explain why there are no answers?
 
Probably easiest to visualize in a Penrose diagram like this one, where all light rays are drawn at 45 degree angles (and all timelike worldlines are closer to vertical than 45 degrees) so a light cone from any event would look just like it does in a Minkowski diagram (compare with Penrose diagram for a nonrotating Schwarzschild black hole here and here). Also, it's not quite what you're asking for but you may be interested in the "waterfall diagrams" on this page and this one, with more details about these types of diagrams given in the paper linked to on the second page.
 
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