Pournelle's Empire of Man: What happened to Sparta's other royal family? (spoilers)

In Jerry Pournelle’s CoDominium/Empire of Man SF universe, the colony world of Sparta has a government founded by political scientists and consciously modeled, in some respects, on that of the ancient Greek city of Sparta. The torturously complex system (in Prince of Sparta, IIRC, even the heir to the throne, Lysander Collins, says at one point, “I’d hate to have to explain it”) includes two kings and two royal families – Collins and I forget the other one. At the end of Go Tell the Spartans, Lysander has succeeded the throne, and, following the defeat of a rebellion of the “helots,” the leading officers of the armed forces of the collapsed CoDominium acclaims him “imperator” of the new Empire of Man.

The Empire of Man endures, collapses, rises again, all under a line of emperors descended from Lysander; but the other Spartan royal family is never mentioned again, to the best of my recollection. Did they just drop the dyarchy at some point?

You could try asking JP: