A generalized Stieltjes system with polynomial source

Let $Q$ be a monic polynomial of degree $M+1$. We study the algebraic system $$\sum _{j\mathrm{\ne }i}\frac{1}{x_i-x_j}=Q(x_i),i=1,\dots ,N,$$ for pairwise distinct complex numbers $x_1,\dots ,x_N$, modulo permutations of these numbers. The case $M=0$ is, after a translation, the classical Stieltjes system for the zeros of a Hermite polynomial. We prove that, for arbitrary $Q$, the number of solutions is at most $\left(\genfrac{}{}{0px}{}{N+M}{N}\right)$, and that the coefficient equations for the associated monic Stieltjes polynomial have total intersection multiplicity exactly $\left(\genfrac{}{}{0px}{}{N+M}{N}\right)$. Consequently the bound is attained for all $Q$ in a non-empty Zariski open subset of the affine space of monic polynomials of degree $M+1$. We also describe the solutions when the coefficient of the linear term of $Q$ is large: the system splits into $M+1$ weakly coupled classical Stieltjes systems, one near each zero of $Q$.