Number of Integer Solutions to Homogeneous System in Sphere

To find the number of integer solutions $(x, y, z)$ satisfying the given system of homogeneous equations:

$$\begin{aligned} 3x - y - z &= 0 \quad &\text{--- (1)} \\ -3x + z &= 0 \quad &\text{--- (2)} \\ -3x + 2y + z &= 0 \quad &\text{--- (3)} \end{aligned}$$

From equation (2), we have:
$z = 3x$

Substituting $z = 3x$ into equation (1):
$3x - y - 3x = 0 \implies -y = 0 \implies y = 0$

Now, substituting $y = 0$ and $z = 3x$ into equation (3):
$-3x + 2(0) + 3x = 0 \implies 0 = 0$
which holds identically.

Thus, the general solution to the system is:
$(x, y, z) = (x, 0, 3x)$

Since $x, y, z$ must be integers, $x$ must be an integer, which automatically ensures that $y = 0$ and $z = 3x$ are also integers.

We are given the inequality:
$x^2 + y^2 + z^2 \le 100$

Substituting $y = 0$ and $z = 3x$ into this condition:
$x^2 + 0^2 + (3x)^2 \le 100$
$10x^2 \le 100$
$x^2 \le 10$

Since $x \in \mathbb{Z}$, the possible values of $x$ are:
$x \in \{-3, -2, -1, 0, 1, 2, 3\}$

There are $7$ possible values for $x$, each corresponding to a unique integer point $(x, 0, 3x)$.

Therefore, the number of such points is $7$.