We study the lattice $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ of subvarieties of the ai-semiring variety $${{\mathbf{CSr}}}(n, 1)$$ defined by $$x^n\approx x$$ and $$xy\approx yx$$ . We divide $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ into five intervals and provide an explicit description of each member of these intervals except $$[{{\mathbf{CSr}}}(2, 1), {\mathbf{CSr}}(n, 1)]$$ . Based on these results, we show that if $$n-1$$ is square-free, then $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ is a distributive lattice of order $$2+2^{r+1}+3^r$$ , where r denotes the number of prime divisors of $$n-1$$ . Also, all members of $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ are finitely based and finitely generated and so $${{\mathbf{CSr}}}(n, 1)$$ is a Cross variety. Moreover, the axiomatic rank of each member of $${\mathscr {L}}({{\mathbf{CSr}}}(n, 1))$$ is less than four.