The XYZ Equations

Algebra Level 5

$\begin{array}{l l} x^2 y^2 + y^2 z^2 = 5 xyz\\ y^2 z^2 + z^2 x^2 = 17 xyz \\ z^2 x^2 + x^2 y^2 = 20 xyz \\ \end{array}$

How many ordered sets of integers $(x, y, z)$ such that $x,y,$ and $z$ are between -10 and 10 inclusive are solutions to the system of equations above?