New Encyclopaedia Britannica, 15th ed., Chicago, 1992, vol. 12, pg. 411 describes Anne as the sister of Basil II. Butler's lives of the saints, NY : P. J. Kennedy & Sons, 1959, pg. 110 describes her as his daughter.
Anna Porphyrogenita (March 13, 963 ? 1011) was a Grand Princess consort of Kiev; she was married to Grand Prince Vladimir the Great.
Anna was the daughter of Byzantine Emperor Romanos II and the Empress Theophano. She was also the sister of Emperors Basil II Bulgaroktonos (The Bulgar-Slayer) and Constantine VIII. Anna was a Porphyrogenita, a legitimate daughter born in the special purple chamber of the Byzantine Emperor's Palace. Anna's hand was considered such a prize that Vladimir became Christian just to marry her.
Anna did not wish to marry Vladimir and expressed deep distress on her way to her wedding. Grand Prince Vladimir was impressed by Byzantine religious practices, this factor, along with his marriage to Anna led to his decision to convert to Eastern Christianity. Due to these two factors, Grand Prince Vladimir also began Christianizing his kingdom. By marriage to Grand Prince Vladimir, Anna became Grand Princess of Kiev, but in practice, she was referred to as Queen or Czarina, probably as a sign of her membership of the Imperial Byzantine House. Anna participated actively in the Christianization of Rus: she acted as the religious adviser of Vladimir and founded a few convents and churches herself. It is not known whether she was the biological mother of any of Vladimir's children, although some scholars have pointed to evidence that she and Vladimir may have had as many as three children together (wikipedia.org)