Okay, I got the solution. While obtaining the characteristics I, in general, used

$\frac{\mathrm{d}y}{\mathrm{d}x}=\pm i\frac{1}{\sqry{y}}$ but here it makes problem.

Thus I took the reciprocal equation $\frac{\mathrm{d}y}{\mathrm{d}x}=\pm i\sqrt{y}$,

which gives $\xi=\frac{3}{2}x+(-y)^{3/2}$ and $\eta=\frac{3}{2}x-(-y)^{3/2}$.

Thus, the equation transforms into $z_{\xi\eta}+\frac{1}{6(\xi-\eta)}(z_{\xi}-z_{\eta})=0$.

Have a good day.

bkarpuz