Commutants of the sum of two quasihomogeneous Toeplitz operators

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is the complete characterization of the set of all Toeplitz operators that commute with a given operator. In \cite{al}, the authors showed that when a sum $S=T_{e^{im\theta}f}+T_{e^{il\theta}g}$, where $f$ and $g$ are radial functions, commutes with a sum $T=T_{e^{ip\theta}r^{(2M+1)p}}+T_{e^{is\theta}r^{(2N+1)s}}$, then $S$ must be of the form $S=cT$, where $c$ is a constant. In this article, we will replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ with $r^n$ and $r^d$, where $n$ and $d$ are in $\mathbb{N}$, and we will show that the same result holds.