Maybe this 2001 Notices of the AMS article (PDF link) by Richard Montgomery is useful: it describes a figure-8 solution to the 3-body problem as well as some other "coreographies". Some of the latter ones were found numerically by Carles Simó.
EDIT: It is probably not the same figure-8 solution mentioned in the question, since the masses of the three bodies are equal in the case of the Chenciner--MontomeryMontgomery solution.