Quadrant uniqueness for unequal dilation characteristics ($x{z}_x+2y{z}_y=xy,z(s,s)=s^2$)

The method of characteristics from $s\ne 0$ are $(x,y,z)=(s{e}^t,s{e}^{2t},s^2(2+e^{3t})/3)$. Their projections are half-parabolas $y=x^2/s$ with $x/s>0$. The inverse parameters $s=x^2/y$, $t=\log (y/x)$ exist exactly on the first and third quadrants, giving the unique solution $z=2x^4/(3y^2)+xy/3$ there. The projection Jacobian $s{e}^{3t}$ is nonzero, so every allowed initial point is noncharacteristic.