Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes

Let $X$ be a nonempty set and $E$ an equivalence relation on $X$. The semigroup $T_E(X)$ consists of all transformations on $X$ that preserve $E$. Although the collection of all regular elements of $T_E(X)$ is not, in general, a subsemigroup, we investigate maximal regular subsemigroups of $T_E(X)$