  Jackson and Lee proved that certain six-element monoid $E^1$ generates a
variety $\mathbb E$ whose lattice of subvarieties contains an infinite
ascending chain $\mathbb E(\sigma_1) \subset \mathbb E(\sigma_2) \subset
\dots$. We show that every subvariety of $\mathbb E$ is generated by a monoid
of the form $M_\tau(W)$ for certain congruence $\tau$ on the free monoid. Using
$M_\tau(W)$ generators for $\mathbb E(\sigma_2)$ and its dual
$\overline{\mathbb E (\sigma_2)}$, we show that these varieties are contained
in three new limit varieties.
