After asking this question and finding this relevant paper, I would like to ask the following question:

For every $a,b \in \mathbb{C}$, denote: $A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$ and $B_{a,b}=\mathbb{C}[(y-a)(y-b),y(y-a)(y-b),x]$.

If I am not wrong (but I may be wrong), $A_{a,b}$ and $B_{a,b}$ are maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$, namely, there is no sub-$\mathbb{C}$-algebra $T_A$ such that $A_{a,b} \subsetneq T_A \subsetneq \mathbb{C}[x,y]$ and there is no sub-$\mathbb{C}$-algebra $T_B$ such that $B_{a,b} \subsetneq T_B \subsetneq \mathbb{C}[x,y]$.

Question: I am looking for maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$ other than $A_{a,b},B_{a,b}$. Is it possible to describe all maximal sub-$\mathbb{C}$-algebra of $\mathbb{C}[x,y]$ in terms of generators like $A_{a,b},B_{a,b}$?

Thank you very much!

After asking this question and finding this relevant paper, I would like to ask the following question:

For every $a,b \in \mathbb{C}$, denote: $A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$ and $B_{a,b}=\mathbb{C}[(y-a)(y-b),y(y-a)(y-b),x]$.

If I am not wrong (but I may be wrong), $A_{a,b}$ and $B_{a,b}$ are maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$, namely, there is no proper sub-$\mathbb{C}$-algebra $T_A$ of $\mathbb{C}[x,y]$ such that $A_{a,b} \subsetneq T_A$ and there is no proper sub-$\mathbb{C}$-algebra $T_B$ of $\mathbb{C}[x,y]$ such that $B_{a,b} \subsetneq T_B$.

Question: I am looking for maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$ other than $A_{a,b},B_{a,b}$. Is it possible to describe all maximal sub-$\mathbb{C}$-algebra of $\mathbb{C}[x,y]$ in terms of generators like $A_{a,b},B_{a,b}$?

Thank you very much!