= Solution
A <characteristic hypersurface> of the form $\phi(t,x)=p(t)\pm q(x)$ must satisfy
$$
tx\,p'(t)^2-q'(x)^2=0.
$$
On either connected component of $tx>0$, separation gives
$$
tp'(t)^2=\frac{q'(x)^2}{x}=\operatorname{sgn}(t)=\operatorname{sgn}(x).
$$
One choice valid on both components is
$$
p(t)=2\operatorname{sgn}(t)\sqrt{|t|},
\qquad
q(x)=\frac23\operatorname{sgn}(x)|x|^{3/2}.
$$
Indeed, $p'(t)=|t|^{-1/2}$ and $q'(x)=|x|^{1/2}$ away from the axes, so $txp'^2=q'^2$. Thus
$$
\boxed{p(t)+q(x)=\text{constant},\qquad p(t)-q(x)=\text{constant}}
$$
are the two families of characteristics, and $p\pm q$ are <characteristic coordinate>[characteristic coordinates].
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