Hope, MO is the right place for this question (if not so: where would you pose it?).
Consider a two-body system in classical mechanics. As long as the interaction depends only on the distance of the two bodies, the two-body problem is integrable/solvable. Now consider the two bodies in a fixed external field. (This is only one step away from a three-body system that is known to be non-integrable in general, but obviously different from it.)
Question: Can the conditions on the combination of interaction and external field be explicitely given for the problem to be integrable/solvable?
It might be the case that the problem is always solvable. In this case the following reference request becomes predominant:
Reference request: Where can I find an explicit and elaborated treatment of this problem?