I would like to solve the following system of equations over the real numbers: $$\left\{ \begin{array}{rcl} x + y^2 + z^3 &=& 3\\ y + z^2 + x^3 &=& 3\\ z + x^2 + y^3 &=& 3\\ \end{array} \right.$$ According to Mathematica there is only one solution, $(x, y, z) = (1, 1, 1)$. I can show that for all other solutions, $x$, $y$ and $z$ would have to be (pairwise) distinct, so I can assume $x < y <z$ or $x < z <y$, but that's all I have so far. I'm mainly interested in real solutions, but I would also like to know if there are any other complex solutions.
Thanks in advance.
$x,y,z$ are roots of the degree $27$ polynomial
$x^{27}-27x^{24}+317x^{21}-18x^{19}-2067x^{18}-50x^{17}+279x^{16}+8156x^{15}+645x^{14}-1674x^{13}-20359x^{12}-3044x^{11}+4645x^{10}+33644x^9 + 6288x^8-6388x^7-36936x^6-5925x^5+4957x^4+23187x^3+4063x^2-4342x-5352$
The triangles show how to pick $(x,z,y)$ to form solutions (I think I messed up the ordering)
$1$ is indeed the only real root.