Finding the least number of equations for which that $S-1$ is a component.

Question

Let $S=\sum w$, summed over all words $w=\{x,y\}$ with the same number of $x$'s and $y$'s. Let $\phi$ be the abelianization operator. We know that

$$\phi(S)=\sum_{n\geq1}{2n\choose n}x^ny^n=\frac{1}{\sqrt{1-4xy}}$$

My question here is: what is the least number of equations in a proper algebraic system for which that $S-1$ is a component?

Answer 1

Hint: Let $Q$ be series: $Q=1+xQyQ+yQxQ$, and therefore $Q$ is algebraic.