Solving a second-order partial differential equation with non-linear first order terms

Question

I am very new in Stack Exchange. I would like to solve the equation $\phi_{xy}+\phi_y^2-\phi_x^2=0$. I read a summary of the method of characteristics in the book of Zwillinger, and there is something which I don't understand. As far as I understand, one writes $v=\phi_x$ and $w=\phi_y$, and we have to solve a system of two equations \begin{align} v_y+w^2-v^2&=0\\ v_y-w_x&=0 \end{align} Do we seek a characteristic like $v(x(s),y(s))$ and $w(x(s),y(s))$ ? With the first equation, we should end up with $$ x'(s)=0 \text{ and } y'(s)=1 \text{ and } dv/ds=v^2-w^2$$ The first two equations are easy to solve and show that the characteristics lines are parallel to $Oy$. The problem I face is : What is $w(x,s)$ in the last expression ? The last equation $v_y-w_x=0$, when compared to $dw/ds$ taken on the characteristics, gives an inconsistency, since the coefficient in front of $w_x$ is $dx/ds=0$.. Could someone tell me how (in principle) to solve $\phi_{xy}+\phi_y^2-\phi_x^2=0$, and where my mistake is ? Thanks a lot..