Determination of the orders generated by a cyclic cubic unit that are Galois invariant

Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field Q(ϵ) is Galois. In this situation, it is natural to ask when the cubic order Z[ϵ] is invariant under the action of the Galois group Gal(Q(ϵ)/Q). It seems that this natural problem has never been looked at. We give an answer to this problem (e.g., we show that if ϵ is totally positive, then this happens in only 12 cases).