JBMO-SL 2013 A1

Find all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:

$$\begin{aligned} x^{3} & =\frac{z}{y}-2 \frac{y}{z} \\ y^{3} & =\frac{x}{z}-2 \frac{z}{x} \\ z^{3} & =\frac{y}{x}-2 \frac{x}{y} \end{aligned}$$