Cyclic System of quadratic equations Find all solutions to system of the equations
$$
\begin{align*}
x^2&=a+y\\
y^2&=a+z\\
z^2&=a+x\\
\end{align*}
$$
I have only found 2 solutions by setting $x=y=z$ but there can be a total of 8 solutions.

we have from the second equation $$z=y^2-a$$ and from the third equation
$$(y^2-a)^2=a+x$$ with $$y=x^2-a$$ we get $$((x^2-a)^2-a)^2=a+x$$ 
finally we obtain $$-x^8+4ax^6-6a^2x^4+4a^3x^2+2ax^4-a^4-4a^2x^2+2a^3-a^2+a+x=0$$
this equation can be factorized into $$ \left( -{x}^{2}+x+a \right)  \left( -{x}^{6}-{x}^{5}+3\,{x}^{4}a-{x}^
{4}+2\,{x}^{3}a-3\,{x}^{2}{a}^{2}-{x}^{3}+3\,{x}^{2}a-x{a}^{2}+{a}^{3}
-{x}^{2}+2\,xa-2\,{a}^{2}-x+a-1 \right) 
$$
