Limit varieties generated by finite non-J-trivial aperiodic monoids

Jackson and Lee proved that certain six-element monoid generates a hereditarily finitely based variety ${\mathbb{E}}^1$ whose lattice of subvarieties contains an infinite ascending chain. We identify syntactic monoids which generate finitely generated subvarieties of ${\mathbb{E}}^1$ and show that one of these finite monoids together with certain seven-element monoid generates a new limit variety.