Sometimes, despite your best Dr. Phil impression, certain problems just can't be solved in a timely matter. In these cases, it's time to try to find a temporary fix, or a "band-aid" solution.
For example, if it is a separated couple arguing over who keeps what, a temporary fix may be to leave everything as it is until the two can see their attorneys the next morning and work it out legally. If both parties agree to this, the problem is solved for the moment. This still frees you up to get back to your other calls and keeps both people happy, however temporary it is.
Lay Down the Law
Last but not least, when all else fails, some people or problems (or combinations of both) just can't be satisfied no matter your effort level. Some people will stick to their guns and not budge an inch, and others seem to be in desperate situations that they simply won't see a way out of.
In these situations, when your best efforts have failed, it's time to tell them what a civil problem really is. It is not a police matter, it is a private matter between two parties that the police department often has no power or authority to rule on. That's what civic court is for, and that is the route these people will likely have to go.