Let $\cal{P}$ be the variety of semigroups defined by the identity $xyzx\approx x^2$. By a result of György Pollák, every subvariety of $\cal{P}$ is finitely based. The present article is concerned with subvarieties of $\cal{P}$ and the lattice they constitute, where the main result is a characterization of finitely generated subvarieties of $\cal{P}$. It is shown that a subvariety of $\cal{P}$ is finitely generated if and only if it contains finitely many subvarieties, and the identities defining these varieties are described. Specifically, it is decidable when a finite set of identities defines a finitely generated subvariety of $\cal{P}$. It follows that the finitely generated subvarieties of $\cal{P}$ constitute an incomplete lattice while the non-finitely generated subvarieties of $\cal{P}$ constitute an interval. It is also shown that given any pair of finitely generated subvarieties of $\cal{P}$, a finite semigroup that generates their meet is computable.