Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (here $\langle \langle ... \rangle \rangle$ stands for normal closure). It is not hard to see, that $L(\mathfrak{U})$ is closed under subgroups, quotients and direct products, and thus is a variety too. So $L$ is an operator on group varieties. If I am not mistaken, it is usually called Levi operator.
It is always true, that $\mathfrak{U} \subset L(\mathfrak{U})$, however the converse is not true in general: for example, Levi operator maps the variety of abelian groups to the variety of $2$-Engel groups. The variety of all groups and the trivial variety are nevertheless preserved by it.
My question is:
Do there exist such varieties $\mathfrak{U}$, such that $L(\mathfrak{U})=\mathfrak{U}$, but $\exists G \notin \mathfrak{U}$ and $\exists H \in \mathfrak{U}$?
How about the variety of groups of a given exponent $e > 1$?