Symmetric Diophantine equation (noch nicht ├╝bersetzt)

Problem 785

Consider the following Diophantine equation: $$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$ where $x$, $y$ and $z$ are positive integers.

Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1 \le x \le y \le z \le N$ and $\gcd(x, y, z) = 1$.

For $N = 10^2$, there are three such solutions - $(1, 7, 16), (8, 9, 39), (11, 21, 72)$. So $S(10^2) = 184$.

Find $S(10^9)$.