= Solution
Use <geometrized units> $G=c=1$ and <metric signature> $(-,+,+,+)$. Put $f=1-2M/r$ in the <Schwarzschild metric>. The <Schwarzschild tortoise coordinate> satisfies $dr_*/dr=f^{-1}$, so
$$
r_*=r+2M\log\left|\frac r{2M}-1\right|,\qquad v=t+r_*.
$$
Substituting $dt=dv-f^{-1}dr$ gives the <ingoing Eddington-Finkelstein coordinates>:
$$
\boxed{ds^2=-f\,dv^2+2\,dv\,dr+r^2(d\theta^2+\sin^2\theta\,d\phi^2).}
$$
The radial <metric tensor> has determinant $-1$ and inverse components $g^{vv}=0$, $g^{vr}=1$, $g^{rr}=f$. Thus it is nondegenerate and analytic at $r=2M$. The same expression defines a <Lorentzian metric> for every $r>0$, extending the exterior across the future <Schwarzschild event horizon> into the <black hole>. It does not include the other exterior or the <white hole> of the full <Kruskal spacetime>. At $r=0$, the <Kretschmann scalar> $R_{abcd}R^{abcd}=48M^2/r^6$ diverges, so this is a <curvature singularity>, not a removable <coordinate singularity>.
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