My great great grandfather was Salvatore Bax from Teano, Campania, Italy or thereabouts. Supposedly he had a brother Lorenzo Bax. And supposedly they came to Italy from somewhere else, most likely in the mid 1800s. The elders in the family used to say that they were "Russe," meaning that they were from "Russia," which could, of course, has changed geopolitically since the 1800s.
Salvatore was excommunicated from the Catholic Church for doing some kind of hands-on-healing, and maybe also for "women-stealing." He was reinstated after a trip to Rome to petition the Pope. Salvatore's son Domenico had hazel eyes and light hair.
Bax is also a Kazak/Kyrgis/Caucuses Mountain area name - it's not exclusively an Anglo-Saxon name. (info found in a book called A Hundred Thousand Fools of God, in a chapter on traveling bard/musician/storytellers with the surname 'Bax')
Lorenzo and his descendants may have settled in Naples, Italy. Salvatore's descendants lived in Teano still live there, but we are not able to contact them.
I also read somewhere that most of the Bax name in Italy is centered near Molise, Bari, etc, and that's because they came over from Albania as political/religious refugees, not wanting to convert to Islam in Albania. They settled in the towns near Bari where they landed.
IF anybody has any insight/corroboration, we'd be grateful to hear it.