  We find a specific coordinate system that goes from the Painlev\'e-Gullstrand
partial extension to the Kruskal-Szekeres maximal extension and thus exhibit
the maximal extension of the Schwarzschild metric in a unified picture. We do
this by adopting two time coordinates, one being the proper time of a
congruence of outgoing timelike geodesics, the other being the proper time of a
congruence of ingoing timelike geodesics, both parameterized by the same energy
per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit
$E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, through such an
integrated description one sees that the Kruskal-Szekeres solution belongs to
this family of extensions parameterized by $E$. Our family of extensions is
different from the Novikov-Lema\^itre family parameterized also by the energy
$E$ of timelike geodesics, with the Novikov extension holding for $0<E<1$ and
being maximal, and the Lema\^itre extension holding for $1\leq E<\infty$ and
being partial, not maximal, and moreover its $E=\infty$ limit evanescing in a
Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime.
