I have noticed that the Liturgy of the Hours usually does not provide a reading if it is not going to be ever used (duh). For example we can commemorate Pope St. Sylvester I on Dec 31, and there is an antiphon for the Benedictus. But there isn't one for the Magnificat, because it would never be used ... because the evening of the 31st is always First Vespers of the Solemnity of Mary, Mother of God.
So why is there a second reading in the Office of Readings for January 2nd? (though mistranslated as "Monday" in the English, in my Latin version it's clearly for January 2nd).
Per my understanding this reading would never be used, since the reading provided in the proper of saints under Jan 2, memorial (obligatory) of SS Basil and Gregory, would be used instead.