The problem is vastly easier with initial positions and velocities constrained to the plane. If they start that way, they can't ever leave. Even if you have a coulomb/newtonian potential, proving stability will be far more straightforward.
I'd love to be surprised, but I suspect closed and collisionless solutions with any out-of-plane components would be newsworthy and immediately publishable.
> closed and collisionless solutions with any out-of-plane components would be newsworthy and immediately publishable.
Trivial example: What about two tiny spaceships (they're small enough that their effect on each other is negligble) orbiting a star, one orbiting in the XY plane the other in the XZ plane?
I'd love to be surprised, but I suspect closed and collisionless solutions with any out-of-plane components would be newsworthy and immediately publishable.