Solving a system of rational functions

Question

Given $c_k \in \mathbb C$ distinct nonzero complex numbers. Is that possible to find a nontrivial solution for $\vec x = (x_1, x_2, \dots, x_n) \in \mathbb C^n$ such that $\vec x$ satisfies the following system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - x_i} \forall k = 1, 2, 3, \dots, n$$