Example: Constant astigmatism equation

$z_{yy}+(1/z{)}_{xx}+2=0.$

H. Baran and M. Marvan, On integrability of Weingarten surfaces: a forgotten class, J. Phys. A: Math. Theor 42 (2009) 404007.

$A=\left(\begin{array}{cc}\frac{1}{8}\frac{(a^2+1)z_x}{az}+\frac{1}{8}\frac{(-a^2+1)z_y}{a}& \frac{1}{4}\frac{(a+1{)}^2\sqrt{z}}{a}\\ \\ \frac{1}{4}\frac{(a-1{)}^2\sqrt{z}}{a}& -\frac{1}{8}\frac{(a^2+1)z_x}{az}-\frac{1}{8}\frac{(-a^2+1)z_y}{a}\end{array}\right),$

$B=\left(\begin{array}{cc}\frac{1}{8}\frac{(-a^2+1)z_x}{a{z}^2}+\frac{1}{8}\frac{(a^2+1)z_y}{az}& \frac{1}{4}\frac{-a^2+1}{a\sqrt{z}}\\ \\ \frac{1}{4}\frac{-a^2+1}{a\sqrt{z}}& -\frac{1}{8}\frac{(-a^2+1)z_x}{a{z}^2}-\frac{1}{8}\frac{(a^2+1)z_y}{az}\end{array}\right)$