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.
 A: Reasoning on the quadratic factor in the decomposition obtained by @Dr. Sonnhard Graubner, we can place into evidence a threshold : for any $a>-\dfrac{1}{4}$ there are always at least two real valued solutions: 
$$(x,y,z)=(u,u,u) \ \ \ \text{and} \  \ \ (x,y,z)=(v,v,v)$$
with $$u=\dfrac{1 - \sqrt{4a+1}}{2} \ \ \ \text{and} \  \ \  v=\dfrac{1 + \sqrt{4a+1}}{2}$$
In the particular case $a=-\dfrac{1}{4}$, these two solutions coalesce in the single solution $$(x,y,z)=(\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2})$$.
On an experimental basis we can set the conjecture:


*

*for $a<-\dfrac{1}{4}$ there are no real solutions.

*for $a>\dfrac{7}{4}$ there are 8 real solutions, and among them, of course, the two ones we have described at the beginning.
(I have had not enough time to spend on the problem! 
This number 8 is not surprising, on a geometrical basis of interpretation (see figure below in the case $a=-1$), due to Bezout's theorem.
In fact, the three equations represent 3 paraboloic cylinders. Consider the intersection of the first two ones: as their equations have degree 2, their intersection curve (C) has degree $2 \times 2=4$ . The intersection of (C) and the third paraboloic cylinder (degree 2, as the others) has degree $4 \times 2 = 8$ (which, evidently, is the degree of the polynomial given by @Dr. Sonnhard Graubnert) But of course, this does not give the number of real solutions.
Wishing that it has helped a little in the understanding of the issue...

A: 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 $$ \left( -{x}^{2}+x+a \right)  \left( -{x}^{6}-{x}^{5}+3\,{x}^{4}a-{x}^
{4}+2\,{x}^{3}a-3\,{x}^{2}{a}^{2}-{x}^{3}+3\,{x}^{2}a-x{a}^{2}+{a}^{3}
-{x}^{2}+2\,xa-2\,{a}^{2}-x+a-1 \right) 
$$
