If an electron moves around a nucleus in an elliptical path, is the moment of inertia of the electron with respect to the nucleus a constant w.r.t time?

I think that both the electron and nucleus must be in motion so as to conserve linear momentum. If I assume that both move in elliptical paths around the COM, then the abovementioned moment of inertia can be a constant only if they move in their respective elliptical trajectories while maintaining constant separation between them. Is such a motion possible? If yes, then how are the ellipses related to each other?