What's the solution to this system of equation? $$
xy^3z^3 = yx^3z^3 = zx^3y^3
$$
Is there a way to solve this system?
I think the answer is 1, but i can't verify my intuition.

Another way of looking at it, clearly $x=0$, or $y=0$, or $z=0$ is a solution.  So, starting with
\begin{alignat*}{3}
         xy^3z^3 &= yx^3z^3 &&= zx^3y^3
\end{alignat*}
And considering the case where $x \ne 0$, $y \ne 0$, $z \ne 0$
\begin{alignat*}{3}
          y^2z^2 &=  x^2z^2 &&= x^2y^2 
\end{alignat*}
Considering each pair of equalities, we get:
\begin{align*}
          y^2z^2 &= x^2z^2 &      x^2z^2 &= x^2y^2     &     y^2z^2 &= x^2y^2 \\ 
\implies       y &= \pm x  &           z &= \pm x      &          z &= \pm x \\ 
\implies       y &= \pm x  &           z &= \pm x
\end{align*}
Hence the solutions are points of the form
$(0,a,b)$, $(a,0,b)$, $(a,b,0)$, or $(a, \pm a, \pm a)$
for $a,b \in \mathbb{R}$.
