...produce different types (champlevé, cloisonné, enamel painting) of enamel dials, for its own and other brands' watches, after the purchase of Donze Candran (a specialist enamel dial maker). I only have their champlevé pictorial dials (Dragon and Serpent) and they are done extremely well. After close inspections under a loupe for many times, I could not find bubbles or noticeable imperfections.
Patek also has in-house capabilities and additionally use independent enamellers (such as Anita Porchet who also does work for VC, Paiget rose dials, J. Dorze Pailonnē dials) on a non-exclusive basis to supplement its own enamel production. Suzanne Rohr, based on an article I read, does contracting work just for Patek for enamel paintings. The 5131s are done by a mother and son team and I am assuming that they are Patek full time employees. The 5131 cloisonné enamel dials do contain some imperfections (and I have seen several of these under a loupe). I think the reason for the 5131 cloisonné imperfections vs none noticed for Ulysse champlevé is that the different techniques result in different outcomes after the firing procedure.
Other brands might have a few enamellers to produce their own pieces but mihgt not have enough capacities like Ulysse's Donze which does contract works for others (such as Vulcain for it Pegasus dials).
Hope this is helpful.
Gordon