How do these crowns fall off? Is it something that happens often? I thought the crown would normally be glued to the stem before installing or so I've seen in videos.
They just work their way loose sometimes through hand setting, sometimes the wrong crown can be fitted with a bigger thread hole than the stem. Thread lock always helps.
I believe this is a case reference from a third-party manufacturer, probably easier to find a similar looking reference (e.g. 135.070) and buy that corresponding crown
Correct. The case was made by Belove & Arienti Watch Case Company (thus the B&A). B&A was part of D&A - DiVincenzo and Arienti who were casemakers for Norman Morris, the US agent for Omega at that time.
The US made cases don't have the same inventory data that the Swill ones do, so finding an actual part number will be problematic.
Just measure stem size/thread and the hole in the case and start the search.
@X350 XJR may be able to offer some info on US cased Omegas?
