It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: Analytic and Modern Tools pp 507-509, Theorem 15.3.4).

When one only assumes that $\gcd(x, y, z)=1$, we also have the solution $(x, y, z) = (3\cdot2^{\frac{r}{2}},2,1)$, for $r=p-3$. But are there any other nonzero integral solutions with none of $(\mid x \mid, \mid y \mid, \mid z \mid)$ being equal to $1$ ?

newtags, as this requires 300 points, cf. mathoverflow.net/help/privileges/create-tags. $\endgroup$ – Stefan Kohl Dec 9 '15 at 22:39