Integer solutions to AGM iteration

Question

Any integer solution to $a^2+b^2=c^2$ also provides an integer solution $x=c$, $y=a$, $z=c+b$, $w=c-b$ to $$agm(x,y)=agm(z,w)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other non-trivial ($z>x>y>w>0$) integer solutions?

Answer 1

Since $\mathrm{agm}(x, -x) = 0$ and $\mathrm{agm}(0,x) = 0$ you get integer solutions $c=0$ and $a,b \in \mathbb{Z}$