Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations $3x - y - z = 0, -3x + z = 0, -3x + 2y + z = 0.$ Then the number of such points for which $x^{2} + y^{2} + z^{2} \le 100$ is ______.

$3x - y - z = 0,\quad -3x + z = 0,\quad -3x + 2y + z = 0$

$x, y, z \in \mathbb{Z}$

$\text{To find: number of points with } x^2 + y^2 + z^2 \le 100$

$\text{Step 1: solve the system} \Rightarrow \text{all integer solutions } (x, y, z)$

$\text{Step 2: keep those with } x^2 + y^2 + z^2 \le 100$

$-3x + z = 0 \Rightarrow z = 3x$

$3x - y - z = 0 \Rightarrow 3x - y - 3x = 0$

$\Rightarrow y = 0$

$-3x + 2y + z = -3x + 0 + 3x = 0\ \checkmark$

$(x, y, z) = (t, 0, 3t),\quad t \in \mathbb{Z}$

$\text{line through the origin, direction } (1, 0, 3)$

$x^2 + y^2 + z^2 \le 100$

$t^2 + 0 + (3t)^2 = 10t^2 \le 100$

$t^2 \le 10 \Rightarrow |t| \le 3$

$t \in \{-3, -2, -1, 0, 1, 2, 3\}$

$\text{Number of values of } t = 7$

$(t, 0, 3t):\ t = -3,\dots,3 \Rightarrow 7 \text{ points}$

$\boxed{7}$