the Vessels of Zeta produce Children of Zeta, who are either recruited in one of the branches that govern the city (bureaucracy, the Temple or the military), or they are distributed amongst those who have need for fresh blood, whether it is someone (or a couple) who cannot bear children, or someone who has the means to take a new member in their family and has an interest in doing so, for instance to learn their trade or help run a business or whatever else... to that effect, prospective parents are evaluated to make sure they are not just trying to get a "free slave" in all but name, and the child that is given to them enters society fully as a member of their family and a Citizen in their own right.
that said, people who can procreate are free to have their children however they wish.
the specifics of the relationship between Zaton and the girls is up to you to determine through the naming of those relationships, which you can do by starting the game and then opening the menu from within the game itself.