The number of all possible values of  $\theta$  , where   $0<\theta <\pi$  for which the system of equations $(y+z)\cos 3\theta =\left(xyz\right)\sin 3\theta$, $x\sin 3\theta =\frac{2\cos 3\theta }{y}+\frac{2\sin 3\theta }{t},\left(xyz\right)\sin 3\theta =(y+2z)\cos 3\theta +y\sin 3\theta$

has a  solution $(x_0,y_0,z_0)$  with  $y_0{z}_0\ne 0$  is

Let  $xyz=t\Rightarrow t\sin 3\theta -y\cos 3\theta -z\cos 3\theta =0$ ....... (1)

$\Rightarrow t\sin 3\theta -2y\sin 3\theta -2z\cos 3\theta =0$ ..................... (2)

$\Rightarrow t\sin 3\theta -y(\cos 3\theta +\sin 3\theta )-2z\cos 3\theta =0$............ (3)

$y_0\cdot z_0\ne 0$ hence homogeneous equation has nontrivial solution.

$\Rightarrow \sin 3\theta \cos 3\theta (\sin 3\theta -\cos 3\theta )=0$ (eliminating from 1, 2, 3)

clearly  $\sin 3\theta \ne 0,\cos 3\theta \ne 0$

$\Rightarrow \theta =\frac{n\pi }{3}+\frac{\pi }{12},n\in I\Rightarrow x=0,\sin 3\theta \ne 0$

$\Rightarrow \theta =\frac{\pi }{12},\frac{5\pi }{12},\frac{9\pi }{12}.$ Hence, three solutions.