Let $$v_i, x ∈ R^n$$ $$ \alpha ∈ R $$ $$ d_i = \|v_i-x\| ∈ R $$ $$ f: \mathbb{R}^{n} \to \mathbb{R} $$
Consider the equation: $$\sum_{i}\frac{f(x)}{(d_i^2+f(x)^2)^{3/2}} = \alpha $$
Is there an analytical solution for $f(x)$?
For clarity, $v_i$ is a finite set (and in my case, typically with up to 10 vectors) so this is a finite summation.
As noted in the comments above, in general this does not have an analytic solution. However, we can do a little bit to narrow down the form of the solution.
First of all, the $x$-dependence is a red herring. Define $y = f(x)$. Then $$ \sum_i \frac{y}{{(d_i^2 + y^2)}^{3/2}} \;=\; \alpha\, . $$ The quantity $y$, were one able to solve for it, would be a function of $\alpha$ and the $d_i$.
Make a change of variables $y = z / \sqrt{\alpha}$. Then this becomes $$ \sum_i \frac{z}{{(c_i^2 + z^2)}^{3/2}} \;=\; 1\, ,\qquad\qquad (1) $$ where $c_i = d_i \sqrt{\alpha}$. If $z = \Phi\left(\{c_i\}\right)$ is the solution of (1), then the solution to the original problem will be $$ y \;=\; \frac{1}{\sqrt{\alpha}} \, \Phi\left(\{d_i\, \sqrt{\alpha}\}\right)\, . $$
This is as far as you can get in general, but at least this tells you something about how the solution scales.