Let $\quad x=k$

$\Rightarrow \quad$ Put in $(1)\;and\;(2)$

$k-2 y+5 z=0$

$-2 k+4 y+z=0$

$z=0, y=\frac{k}{2}$

$\therefore \quad x , y , z$ are integer

$\Rightarrow \quad k$ is even integer

Now $x=k, y=\frac{k}{2}, z=0$ put in condition

$\begin{array}{l}15 \leq k^{2}+\left(\frac{k}{2}\right)^{2}+0 \leq 150 \\12 \leq k^{2} \leq 120\end{array}$

$\Rightarrow \quad k =\pm 4,\pm 6,\pm 8,\pm 10$

$\Rightarrow$ Number of element in $S =8$