Not the same as Irene of Alania

http://fmg.ac/Projects/MedLands/BYZANTIUM.htm#KonstantinosIXdied1055

Constantine IX Monomachos, byzantine emperor Mistress (2): (after 1044) GORANDUXT of Georgia, daughter of GIORGI I King of Georgia & [his wife Mariam of Vaspurakan]. ... ...

https://fr.m.wikipedia.org/wiki/Constantin_IX#Unions_et_post%C3%A9r...

After the disappearances of Maria Skleraina and the Empress Zoe in 1050 , Constantine IX fell in love in 1054 with the Georgian Princess Gorandouxt, henceforth called Irene, sister of King Bagrat IV of Georgia , hostage in Constantinople, who became his mistress [ 98 ] .

• from Wikipedia, the free encyclopedia

Gurandukht (Georgian: გურანდუხტი) (died before 1072) was a daughter of King George I of Georgia by his first wife Mariam. She was active in the politics of Georgia during the reign of her brother Bagrat IV.[1]

During Bagrat's exile at the Byzantine court enforced by the rebellion of Liparit IV, Duke of Kldekari, in the 1050s, Gurandukht was "protector" (patroni) of Bagrat's young son George II, who was declared king in Bagrat's absence at Constantinople. It was she who demanded that the emperor Constantine IX Monomachos return Bagrat to Georgia.[1] Gurandukht then took part in the restoration of Bagrat's authority in Georgia following the downfall of Liparit.[2]

According to the Chronicle of Kartli, part of the Georgian Chronicles, Gurandukht was married to Smbat, brother of Kyurik II, an Armenian king of Lori. Their unnamed daughter was intended by Bagrat to marry the Seljuk sultan Alp Arslan as part of a peace deal after the sultan's Georgian campaign in 1064, but he failed to obtain consent from his Armenian in-law Kyurik. In the ensuing conflict, Kyurik was captured by Bagrat's agents and forced into submission of the fortress of Samshvilde to the king of Georgia. Gurandukht's daughter was eventually given in marriage to Alp Arslan,[2] but later became the wife of the influential Seljuk vizier and scholar Nizam al-Mulk, with whom she had the son Diya al-Mulk, the future vizier. The circumstances in which she came into Nizam's harem are obscure.[3]

http://en.wikipedia.org/wiki/Constantine_IX_Monomachos#Family

(Constantine IX's)

Family

Constantine Monomachos was married three times:

1. to a wife of unknown identity.
2. to Helena Skleraina, daughter of Basil Skleros, great-granddaughter of Bardas Skleros, and niece of Emperor Romanus III.
3. to the Empress Zoe

• After the death of his second wife, Constantine also took her first cousin Maria Skleraina as his mistress.
• At the time of Constantine's death in January 1055, the emperor had another mistress, a certain "Alan princess", probably Irene, daughter of the Georgian Bagratid prince Demetrius.[33]

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From https://ru.m.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BD%D0%BE%D0%BC%D0%B... retrieved January 2021

After the death of Helena Sklirena, the emperor Constantine took her cousin Maria Sklirena (d. 1044) as his mistress and settled her in the imperial palace in parallel [8] with his marriage to Zoe. Also known is his mistress "Alan" princess - perhaps Irina, daughter of the Bagratid prince Dmitry (d. 1042, son of George I ).

Disputed Origins

Lynda Garland:

http://www.roman-emperors.org/annadal.htm

It is possible that this cousin Irene was the 'Alan princess', a hostage at court who had been mistress of Zoe's third husband Constantine IX Monomachus.Gorn Vilaidaraga In this case she would have been in Constantinople since c. 1054. However, Isaac Comnenus' wife was still producing children in 1096 and it is therefore unlikely that she had been Constantine's mistress more than 40 years earlier. It would also seem unlikely that Anna Dalassena would have sanctioned her son's marriage to an ex-imperial mistress, though in the interest of the family she might have been prepared to overlook an earlier palace liaison.

References

1. Garland, Lynda & Rapp, Stephen (2006), "Mary 'of Alania': Woman and Empress Between Two Worlds", p. 120. In: Garland, Lynda (ed., 2006), Byzantine Women: Varieties of Experience, 800-1200. Ashgate Publishing, ISBN 0-7546-5737-X.
2. Thomson, Robert W. (1996), Rewriting Caucasian History, pp. 296-299. Oxford University Press, ISBN 0-19-826373-2
3. Lambton, Ann K.S. (1988), Continuity and change in medieval Persia, p. 302. SUNY Press, ISBN 0-88706-133-8.