Unfortunately the highly touted "Official British Monarchy" site (https://www.royal.uk) has been so jazzed up and dumbed down that it no longer contains any reference whatsoever to Agatha, Edward the Exile, or his *or* her ancestry.
That puts us back to square one with the Usual Suspects: (from http://sbaldw.home.mindspring.com/hproject/prov/agath000.htm)
The Theories
The main hypotheses are listed here, along with the labels that they have been assigned for purposes of the discussion below. A few impossible theories which can be easily dismissed are not given labels.
The German Hypothesis (main version):
Conjectured father (possible): Liudolf, d. 15 or 23 April 1038, count (Braunschweig).
Conjectured mother (possible): Gertrude.
The Russian Hypothesis:
Conjectured father (possible): Iaroslav I, d. 1054, grand prince of Kiev.
Conjectured mother (possible): Ingegerd, daughter of Olaf, king of Sweden.
The Polish Hypothesis:
Conjectured father (improbable): Mieszko II Lambert, d. 10 May 1034, king of Poland.
Conjectured mother (improbable): Richenza, daughter of Ezzo, count palatine of Lorraine.
The Bulgarian Hypothesis:
Conjectured father (improbable): Gavril Radomir, d. 1015, emperor of Bulgaria.
Conjectured mother (improbable): NN, sister of István (Stephen) I, king of Hungary.
The Hungarian Hypothesis:
Conjectured father (very improbable [but adored by the Hungarians]): István (Stephen) I, d. 1038, king of Hungary.
Conjectured mother (improbable): Gisela, sister of Heinrich II, emperor.
The Cristinus Hypothesis:
Conjectured father (very improbable): Christinus, count.
Conjectured mother (highly improbable): Oda, daughter of Bernhard, count of Haldensleben.
The German Hypothesis (alternate version):
Conjectured father (very improbable): Ernst II, d. 17 August 1030, duke of Swabia.
The Bruno Hypothesis:
Conjectured father (very improbable): Bruno, d. 24 April 1029, bishop of Augsburg, 1007-1029, brother of emperor Heinrich II.
The Byzantine Hypothesis:
Conjectured father (no reasonable basis): Constantine IX "Monomachos", d. 1055, Byzantine emperor.
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It should be noted that out of the entire lineup, the only one known for certain to have daughters in excess (five, of whom three, and their husbands, are known) is Yaroslav of Kiev. This is a point in his favor, but the early insistence on Germany or Hungary for Agatha's origin is a point against.