Analytic solution to system of coupled nonlinear differential equations

Question

Consider the $n$-dimensional dynamical system for $\mathbf{x}=(x_1,...,x_n)$ given by \begin{equation} \dot{x}_i=r_i\left(\sqrt{\frac{\sum_{j\neq i}x_j}{c_i}}-\sum_{j=1}^{n}x_j\right), \end{equation} such that $c_i,r_i\in\mathbb{R}_{>0}$ and $x_i(t)> 0$ for all $t$ with $x_i(0)=A_i>0$. In two-dimensions, this reduces to the following system of nonlinear ODEs: \begin{align} \dot{x}_1&=r_1\left(\sqrt{\frac{x_2}{c_1}}-x_1-x_2\right),\\ \dot{x}_2&=r_2\left(\sqrt{\frac{x_1}{c_2}}-x_1-x_2\right). \end{align} Is there a closed form or power series solution for $(x_1,x_2)$? What about $\mathbf{x}=(x_1,...,x_n)$? To ensure that the constraint $x_i(t)>0$ is met, I have found that $c_i$ should be small. Note that the steady states are \begin{align} x_1^{*}&=\frac{c_2}{(c_1+c_2)^2},\\ x_2^{*}&=\frac{c_1}{(c_1+c_2)^2}, \end{align} for the two-dimensional system and \begin{align} x_1^{*}&=\frac{2\left(c_{2}+c_{3}-c_{1}\right)}{\left(c_{1}+c_{2}+c_{3}\right)^{2}},\\ x_2^{*}&=\frac{2\left(c_{1}+c_{3}-c_{2}\right)}{\left(c_{1}+c_{2}+c_{3}\right)^{2}},\\ x_3^{*}&=\frac{2\left(c_{1}+c_{2}-c_{3}\right)}{\left(c_{1}+c_{2}+c_{3}\right)^{2}}, \end{align} for the three-dimensional system. I haven't found the steady state for $n$-dimensions but one could conjecture $\left(\sum_{j}c_j\right)^2$ to be the denominator. Refer to the plot below for a numerical solution.