[bvwlqjlz] Nice solutions to the three body problem

Create a display showing three bodies under Newtonian gravitation.

What are the set of "nice" solutions to the three body problem?  "Nice" means "bodies stay on screen", and simple (Euler?) numerical integration is sufficient to simulate it stably (though maybe not accurately).  I think the technical terms might be "quasiperiodic" and "stable".

The three body problem is complex, so we're cheating and asking for only the easy solutions.  If we fully simulate three dimensions, do some two dimensional stable solutions (only stable on the "knife edge" of all bodies coplanar) become excluded?

I count 7 degrees of freedom per particle (initial position, velocity, mass), 21 total.  Zero total momentum eliminates 3.  Constant total mass eliminates 1.  First particle must start at origin eliminates 3.  Second particle must start at (1,0,0) eliminates 3.  Third particle must start on the xy plane eliminates 1.

This leaves a 10 dimensional space.  Can it be automatically explored and the "nice" cases automatically classified?  Connected "islands" on the Poincare diagram.  A tricky issue is some of the famous quasiperiodic solutions (Sun Earth Moon, Trojan asteroids, Proxima Centauri) only work when the masses and/or distances are hugely different, so we cannot just uniformly sample over the initial state space. 

Even an incomplete catalog would be fun to watch; add to it as more are discovered.