Let $\alpha$ and $\beta_i$ with $i=1,...,N$ be known real constants, with $0<\alpha<1$ and $0<\beta_i<1$ for all $i=1,...,N$. I need to solve the system of $N+1$ equations defined as
$$ \left\{\begin{array}{rll} z\,\left(1-x_1\,x_2\,\cdots x_N\right)+x_1\,x_2\,\cdots x_N &=& \alpha\\ z\,\left(1-x_1\right)+x_1 & = & \beta_1 \\ z\,\left(1-x_2\right)+x_2 & = & \beta_2 \\ & \vdots & \\ z\,\left(1-x_N\right)+x_N & = & \beta_N \end{array} \right., $$
where $z,x_1,...,x_N$ are the $N+1$-unknowns.
I know that for $N=2$ there is a unique solution given by (verified also with software)
$$ \left\{\begin{array}{rll} z &=& \frac{\alpha-\beta_1\,\beta_2}{1+\alpha-\beta_1-\beta_2} \\ x_1 &=& \frac{\alpha-\beta_1}{\beta_2-1}\\ x_2 &=& \frac{\alpha-\beta_2}{\beta_1-1} \end{array} \right., $$ But at the present time I didn't go any further and I suspect there is no way to find an analytical solution or that there could be more than one solutions. A simple substitution gives that $z$ must solve the equation
$$ \left(\beta_1-z\right)\,\left(\beta_2-z\right)\cdots\left(\beta_N-z\right) = \left(\alpha-z\right)\,\left(1-z\right)^{N-1} $$
so $z$ must one of the roots of a polynomial of degree $N$, which however it does not seem to give any help. Any suggestion is welcome.