Let $\sigma_{ij}, \ i,j=1,\ldots,n$ ($n \geq 4$) be a sequence of positive real numbers such that $\sigma_{ij}=\sigma_{ji}$. Do you know any sufficient condition on $\sigma_{ij}$ (which is simpler than the system itself) such that the linear system of equations
$$ a_i+a_j=\sigma_{ij},\ i,j=1,\ldots,n, i \neq j $$
admits at least a solution? Thank you.