physics - Two interacting bodies in an external field

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?