Let $F$ be a positive rational number, and $j,\ell,k\in \mathbb{Z}^2 \setminus \{(0,0)\}$ with $j=(j_1,j_2)$, $\ell=(\ell_1,\ell_2)$, $k=(k_1,k_2)$.
We suppose that $F$ and $k$ are fixed. How to solve this nonlinear system with respect to $(j,l)$?
$$ \left\{ \begin{align} \dfrac{j_1}{j_1^2+j_2^2+F}+\dfrac{\ell_1}{\ell_1^2+\ell_2^2+F} & = \dfrac{k_1}{k_1^2+k_2^2+F}, \\[8pt] j_1+\ell_1 & =k_1, \\[8pt] j_2+\ell_2 & =k_2. \end{align} \right. $$