The Byzantine presence was the result of a treaty between Justinian and Atanagildo. We know this because they then tried to recover their territories Recaredo asking the Pope he found that treaty, but he could not find the document because it had burned the imperial archive.
Atanagildo (554-567) tried to reclaim the area occupied by the Byzantines attacking troops managed to recover some cities, but not achieved a great success. So now in addition to the already existing problem of Hispano (especially in Seville and Cordoba) in the south is going to unite the problem of the Byzantines.
Atanagildo settled in Toledo as central to dominate the rest of the peninsula, plus two of his daughters, and Galswinta Brunegilda married Frankish princes, which was trying to secure peace with the dangerous northern neighbor. In 567 Atanagildo dies naturally.
From this date until the 590 we will have a lot of data thanks to the Chronicle of John of Biclaro, we also have data by Gregory of Tours, in addition to the acts of the councils, episcopal sources (letters between different bishops) and hagiographical sources (lives of saints).
Five months after the death of King Atanagildo was elected Liuva (567-572), who was Duke of Septimania. But it was not accepted by the Visigoths Hispanic area, so there was a civil war and Liuva after reaching an agreement had to govern only in Septimania and Leovigildo was associated to the throne in Spain.