There is this:
Arthur Zuckerman maintains that Makhir was actually identical with Natronai ben Habibi, an exilarch deposed and exiled...in the late eighth century. Zuckerman further proposed that Makhir(/Natronai) is to be identified with a Maghario, Count of Narbonne.... [and a] William I, Count of Toulouse led Frankish forces at the fall of Barcelona in 803. The account of the campaign in Ermold Niger's Latin poem dates the events according to the Jewish calendar and portrays William as an observant Jew. Count William was son of a Frankish Count of Septimania named Theoderic, leading Zuckerman to conclude that Theoderic was none other than Makhir, and that the well-documented descendants of Theoderic embodied a dynasty of Franco-Judeic kings of Narbonne, representing the union of the lineage of the exilarchs with that of Martel's Carolingians...However, this underlying chain of identifications has been shown to be flawed, a negative opinion shared by other scholars, while the broader suggestions of a Jewish principality in Southern France have likewise been refuted.Where is that?
(Thanks to BT)