3. In the tetrahedron $P A B C$, it is known that
$$
\angle A P B=\angle B P C=\angle C P A=90^{\circ} \text {, }
$$

the sum of the lengths of all edges is $S$. Then the maximum volume of this tetrahedron is $\qquad$ .
3. $\frac{S^{3}}{162(1+\sqrt{2})^{3}}$.

When the edges $A P=B P=C P$ of the tetrahedron $P A B C$ are equal, the volume of the tetrahedron is maximized.
Let $A P=B P=C P=x$. Then
$$
A B=B C=C A=\sqrt{2} x \text {. }
$$

From the given information,
$$
\begin{array}{l}
A P+B P+C P+A B+B C+C A \\
=3(1+\sqrt{2}) x=S .
\end{array}
$$

Thus, $x=\frac{S}{3(1+\sqrt{2})}$.
Therefore, the maximum volume of the tetrahedron $P A B C$ is
$$
\frac{1}{6} x^{3}=\frac{S^{3}}{162(1+\sqrt{2})^{3}} \text {. }
$$