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For $\beta =0$, the equation for the characteristic curves isThe two families are therefore $y=\text{constant}$ and the characteristic projections from part (a), $y=K{x}^2$, together with the vertical characteristic $x=0$. On either region $y>0$ or $y<0$, the hyperbolic partial differential equation admits the characteristic coordinatesIndeed $\phi$ is constant along the first family, while $(x{\partial }_x+2y{\partial }_y)\psi =0$. Their Jacobian determinant isThe coefficients of the two pure second partial derivatives in these coordinates vanish, and the coefficient of the mixed partial derivative is $2y/\sqrt{|y|}\ne 0$. Hence they give the hyperbolic characteristic normal form. They also remain regular at $x=0$, provided $y\ne 0$.

The alternative $\phi =y$, $\psi =y/x^2$ is valid on patches with $xy\ne 0$. On $y=0$, it is a parabolic partial differential equation away from the fully degenerate origin, so two independent hyperbolic characteristic coordinates are unavailable.