An integral system of Matukuma type with negative exponents

We study positive solutions to the integral system$ \begin{cases} u(x) = \int\limits_{ \mathbb R^N}\frac{K(y)v^{-p}(y)}{ |x-y|^{N-\alpha}} dy\quad\text{ for all }x\in \mathbb R^N, \\ v(x) = \int\limits_{ \mathbb R^N}\frac{L(y)u^{-q}(y)}{ |x-y|^{N-\beta}} dy\quad\text{ for all }x\in \mathbb R^N, \end{cases} $where $ p, q > 0 $, $ \alpha, \beta\in (0, N) $ and $ K, L: \mathbb R^N\to (0, \infty) $ are continuous functions which satisfy $ C_1(1+|x|)^{-\gamma}\leq K(x), L(x)\leq C_2(1+|x|)^{-\gamma} $ in $ \mathbb R^N $, for some $ \gamma > 0 $ and constants $ C_2 > C_1 > 0 $. We discuss the existence, nonexistence, and uniqueness of positive solutions to the above system with respect to $ \alpha, \beta, p, q, $ and $ \gamma $. We also classify the finite and infinite total mass solutions of the system.