Ordered triples that sastify the system.

Find all the real system solutions:

$$2x+x^2y=y$$
$$2y+y^2z=z$$
$$2z+z^2x=x$$

My attempt

Trivial solution : (0,0,0)

We have:

$$frac{1}{y}=frac{1-x^2}{2x}$$

$$frac{1}{z}=frac{1-y^2}{2y}$$

$$frac{1}{x}=frac{1-z^2}{2z}$$

Then:

$$x+frac{1}{x} + y+frac{1}{y} + z+frac{1}{z}=0$$

How to procede?