34. We use Pythagorean triples to construct. Take any Pythagorean triple $(x, y, z)$ (not necessarily primitive), and let
$$a=x\left|4 y^{2}-z^{2}\right|, b=y\left|4 x^{2}-z^{2}\right|, c=4 x y z,$$

then we have
$$\begin{aligned} a^{2}+b^{2} & =x^{2}\left(3 y^{2}-x^{2}\right)^{2}+y^{2}\left(3 x^{2}-y^{2}\right)^{2} \\ & =x^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+y^{6} \\ & =\left(x^{2}+y^{2}\right)^{3}=\left(z^{3}\right)^{2}, \\ & a^{2}+c^{2}=x^{2}\left(4 y^{2}+z^{2}\right)^{2}, \\ & b^{2}+c^{2}=y^{2}\left(4 x^{2}+z^{2}\right)^{2} . \end{aligned}$$

Since there are infinitely many Pythagorean triples, there are infinitely many sets of positive integer triples that satisfy the conditions.
For example: when $x=3, y=4, z=5$, we get
$$a=117, b=44, c=240,$$
$$117^{2}+44^{2}=125^{2}, 117^{2}+240^{2}=267^{2}, 44^{2}+240^{2}=244^{2} .$$