Some solution for a particular system of non-linear equations

I have come across this system of equations where $q>1$ is a real constant, $x_i$ and $y_i$ are real variables and $y_i>0$:

\begin{align} 1~~ & = x_0 ~~~~ ~~+ x_1 ~~~~~~~+ x_2 \\ q~~ & = x_0 y_0 ~~+ x_1 y_1 ~~~+ x_2 y_2 \\ q~~ & = x_0 y_0^{-1} + x_1 y_1^{-1} + x_2 y_2^{-1} \\ q^4 & = x_0 y_0^2 ~~+ x_1 y_1^2 ~~~+ x_2 y_2^2 \\ q^4 & = x_0 y_0^{-2}+ x_1 y_1^{-2} + x_2 y_2^{-2} \\ \end{align}

or equivalently

\begin{align} q~~ &= y_0 + x_1 (y_1 - y_0) + x_2 (y_2 - y_0) \\ q^4 &= y_0^2 + x_1 (y_1^2 - y_0^2) + x_2 (y_2^2 - y_0^2) \\ q \cdot y_0 y_1 y_2 &= y_1 y_2 + x_1 y_2 (y_0 -y_1) + x_2 y_1 (y_0 - y_2) \\ q^{4} \cdot (y_0 y_1 y_2)^2 &= (y_1 y_2)^2 + x_1 y_2^2 (y_0^2 -y_1^2) + x_2 y_1^2 (y_0^2 - y_2^2) \end{align}

I am interested in finding a solution to the equations. Any ideas?

• I hope that you noticed that you show five equations for six unknowns. – Claude Leibovici Mar 9 '14 at 19:16
• Indeed, and I was hoping that this would increase my chance to find some solution. – rerx Mar 9 '14 at 19:43