I guess I misunderstood or mis-scoped the problem. Does the 3-body problem state 'the general case' has no solution, but that does not preclude some configurations from having a solution?
It's also not computable, as in chaotic. Small differences in initial positions will lead to unpredictably large differences in trajectory (with small and large having specific meanings to match the formal definition of a chaotic system).
Very cool visuals and site!
Could I make a suggestion:
You show the masses (1,1,1), but not the starting positions, which alter the course of events too.
Why would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?
(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)
(but yes I think you might be right if we're centered on the CG)