Consider the following polynomial equation on the variables $x_1, \ldots, x_n$
$$
\frac{\frac{d}{dz}Q(z)}{ \deg(Q(z))\ q(z)} = z^{n-1},\\
q(z) := (z - x_1)^{k_1}(z - x_2)^{k_2}\ldots (z - x_n)^{k_n},\\
Q(z) := (z - x_1)^{k_1 + 1}(z - x_2)^{k_2 + 1}\ldots (z - x_n)^{k_n + 1},\\
x_1,\ldots,x_n \in \mathbb C\setminus\{0\}\\
x_1,\ldots,x_n \text{ are pairwise distinct}
$$
Notice that I'm only interested in finding at least one solution of the equation.
If we denote $h(z) := (z - x_1)(z - x_2)\ldots (z - x_n)$
and then simplify the left of the equation part we obtain
$$
\frac{1}{\deg((Q(z))}\sum\limits_{s=1}^{n}\frac{k_s + 1}{z - x_s}h(z) = z^{n-1}
$$
That then produces the following system of equations
$$
\begin{cases}
0 = c_1 x_1 + c_2 x_2 + \ldots + c_n x_n\\
0 = \sum\limits_{1 \leq s_1 < s_2 \leq n} c_{s_1, s_2} x_{s_1}x_{s_2}\\
\ldots\\
0 = \sum\limits_{1 \leq s_1 < s_2 < \ldots < s_{n-1} \leq n} c_{s_1, s_2, \ldots, s_{n-1}} x_{s_1}x_{s_2}\ldots x_{s_{n-1}}\\
\end{cases}
$$
where coefficients $c_{*}$ can be linearly expressed with $k_1, \ldots, k_n.$
If $k_1 = \ldots = k_n$ then the system is equivalent to $$ 0 = e_1(x_1, \ldots, x_n) = \ldots = e_{n-1}(x_1, \ldots, x_n), $$ where $e_s(x_1, \ldots, x_n)$ denotes the elementary symmetric polynomial of the degree $s$. The solution to that simplified case are the zeros of the polynomial $z^n - c$ for some $c\in \mathbb C\setminus\{0\}$.
However for the general case it's not clear how to constructively find a solution. Is there any research available on these kinds of, what I've called in the title, elementary semi-symmetric polynomials? Or perhaps I am missing something simpler here?
Any help is really appreciated.