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I am faced with the following system of equations and I'm looking for tools that allow me to characterize its solutions. 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. The $\sigma_{ij}$ 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:

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

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

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

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