Find all real solutions for the system: $x^3=y+y^5$, $y^5=z+z^7$, $z^7=x+x^3.$ 
Given: $$\left\{
\begin{array}{l}
x^3=y+y^5\\ 
y^5=z+z^7\\
z^7=x+x^3 
\end{array}
\right. 
$$
  Find: all real solutions for the system.

From a book on preparation for math contests. The answer states there is just one solution. My problem is showing that this is indeed the case.
My attempt: it is easy to see that one solution is $(x,y,z)=(0,0,0)$. And, adding the equations, we get to $x+y+z=0$. But my problem is showing that this solution is indeed unique, if the answer provided in the book is right. 
Hints and answers are appreciated. Sorry if this is a duplicate. 

Let $x>0$.
Hence, $z^7=x(1+x^2)>0$, which gives $z>0$.
Also, $y^5=z(1+z^6)>0$, which gives $y>0$.
But summing of all equations gives $x+y+z=0$, which is a contradiction.
By the same way we can get a contradiction for $x<0$.
Thus, $x=0$ and from here we obtain $x=y=z=0.$
