$CEA$ operators and the Ershov hierarchy

We examine the relationship between the $CEA$ hierarchy and the Ershov hierarchy within $\Delta_2^0$ Turing degrees. We study the long-standing problem raised in [1] about the existence of a low computably enumerable (c.e.) degree $\mathbf{a}$ for which the class of all non-c.e. $CEA(\mathbf{a})$ degrees does not contain $2$-c.e. degrees. We solve the problem by proving a stronger result: there exists a noncomputable low c.e. degree $\mathbf{a}$ such that any $CEA(\mathbf{a})$ $\omega$-c.e. degree is c.e. Also we discuss related questions and possible generalizations of this result.