I am faced with the following system of equations and I'm looking for tools that allow me to characterize the solution. The unknowns are $\{x_i\}$, $\{y_i\}$, and $\lambda$, for $i=1,...,N$. So there are $2N+1$ equations and the same number of unknowns. Note the subscript for the summations. $\sigma_{ij}$ are entries of a full-rank, invertible $N$ by $N$ matrix.

If helpful, I know the following properties of the parameters:

$\sum_i \beta_i =1$

$\sum_j \sigma_{ij} = 1-\alpha_i >0 $

The system of equations I want to solve is the following:

$x_i +\alpha_i*\lambda = \sum_j \sigma_{ij} y_j$

$\frac{1}{x_j} - \beta_j = \sum_j \sigma_{ij} \frac{1}{y_j}$

$\sum_i \frac{1}{x_i} = \sum_i \frac{1}{y_i}$