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Basil II the Bulgar-Slayer was one of the greatest Byzantine emperors, but he never had an heir. What if he did?
Three variations of this scenario:
Emperor Basil marries in the early 1010s and has a legitimate son, who succeeds him in 1025.
Emperor Basil marries in the early 1010s and has a legitimate son, who succeeds him in the 1030s, as the emperor lives longer.
An illegitimate son of Basil from before his ascent to emperor, shows up in the 1020s with a reasonable case for his descent from Basil. The emperor adopts him as his successor.