Example: Constant astigmatism equation

Equation

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

[B-M]

Zero curvature representation

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

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

[B-M]

References

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