Partial differential equation - integral surface

$x(y^2+z)p - y(x^2+z)q = (x^2-y^2)z$ is ambiguous because the symbols are not defined. I suppose that the PDE is : $$x(y^2+z)\frac{\partial z}{\partial x} - y(x^2+z)\frac{\partial z}{\partial y} = (x^2-y^2)z(x,y)\tag 1$$ The Charpit-Lagrange characteristic ODEs are : $$\frac{dx}{x(y^2+z)}=\frac{dy}{-y(x^2+z)}=\frac{dz}{(x^2-y^2)z}$$ Search for a first characteristic equation : $$\frac{ydx+xdy}{y(x(y^2+z))+x(-y(x^2+z))}=\frac{dz}{(x^2-y^2)z}=\frac{d(xy)}{xy(y^2-x^2)}$$ $$\frac{dz}{z}=-\frac{d(xy)}{xy}$$ $$xyz=c_1$$ Seach for a second characteristic equation : $$\frac{xdx+ydy}{x(x(y^2+z))+y(-y(x^2+z))}=\frac{dz}{(x^2-y^2)z}=\frac{d(x^2+y^2)/2}{(x^2-y^2)z}$$ $$dz=d(x^2+y^2)/2$$ $$z-\frac12 (x^2+y^2)=c_2$$ The general solution of the PDE on implicit form $c_2=F(c_1)$ is : $$z-\frac12 (x^2+y^2)=F(xyz)\tag 2$$ $F$ is an arbitrary function (to be determined according to the boundary condition).

Condition : $z=1$ on $y=-x$ $$1-\frac12 (x^2+(-x)^2)=F(x(-x))$$ $$F(-x^2)=1-x^2$$ Let $X=-x^2$ $$F(X)=1+X$$ Now the function $F$ is determined. We put it into the general solution Eq.$(2)$ where $X=xyz$ $$z-\frac12 (x^2+y^2)=1+xyz$$ $$z(x,y)=\frac{1+\frac12 (x^2+y^2)}{1-xy}$$