= Solution
Use <geometrized units> and <metric signature> $(-+++)$, with $M>0$. In the exterior of <Schwarzschild spacetime>, put $f=1-2M/r$. The <Schwarzschild tortoise coordinate> satisfies
$$
\frac{dr_*}{dr}=f^{-1},\qquad r_*=r+2M\log\left|\frac r{2M}-1\right|.
$$
The <retarded and advanced null coordinates> $u=t-r_*$ and $v=t+r_*$ then give $ds^2=-f\,du\,dv+r^2d\Omega^2$. The logarithmic divergence of $r_*$ suggests exponentiating these <null coordinates>. In the right exterior define the <Kruskal–Szekeres coordinates>
$$
U=-e^{-u/(4M)},\qquad V=e^{v/(4M)}.
$$
Their product eliminates $t$:
$$
UV=-e^{r_*/(2M)}=\left(1-\frac r{2M}\right)e^{r/(2M)}.
$$
Since $du=-4M\,dU/U$ and $dv=4M\,dV/V$, the <Schwarzschild metric> becomes
$$
\boxed{ds^2=-\frac{32M^3}{r}e^{-r/(2M)}\,dU\,dV+r^2d\Omega^2,\qquad UV=\left(1-\frac r{2M}\right)e^{r/(2M)}.}
$$
Here $r$ is an implicitly defined function of $UV$. The derivative of the right side with respect to $r$ is $-r e^{r/(2M)}/(4M^2)$, which is nonzero at $r=2M$. The <inverse function theorem> therefore makes $r(UV)$ smooth across that surface, and the coefficient of $dU\,dV$ tends to $-16M^2/e$. Thus the <Schwarzschild event horizon> is a coordinate singularity of the original chart, while this <Lorentzian metric> remains regular there.
Extend the <Kruskal–Szekeres coordinates> to all real $U,V$ with $UV<1$. The signs give two exterior regions, $U<0<V$ and $V<0<U$, a future <black hole> region $U,V>0$, and a past <white hole> region $U,V<0$. The <event horizons> are $U=0$ or $V=0$, intersecting at the <bifurcation surface>. The boundary $UV=1$ has $r=0$ and is a genuine <Schwarzschild singularity>, as the <Kretschmann scalar> $48M^2/r^6$ diverges there.
Finally, $T=(V+U)/2$ and $X=(V-U)/2$ give $UV=T^2-X^2$ and a radial metric proportional to $-dT^2+dX^2$. Hence radial <null geodesics> have slopes $\pm1$, the <event horizons> are $T=\pm X$, and the singular boundaries are $T^2-X^2=1$. \b[This constructs the maximal Kruskal extension]; a <black hole> produced by collapse need not contain the second exterior or the <white hole> of that eternal extension.
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