name for 2D Electrostatics as Integrable System

I am trying to understand 2D electrostatics of $n$ point charges. Roughly,

$$H = \sum_{i=1}^N n_i \ln |z- z_i|$$

However, I keep bumping across the Gaudin model instead with this Hamiltonian

$$H_j = \sum_{i \neq j} \frac{n_i}{z_j- z_i}$$

and these are often put into context with other integrable systems, such as Calogero-Moser, or Ruisenaars-Schneider. They talk about how $[H_i, H_j] = 0$ as operators and a torus fibration of the phase space.

Is there a larger family of integrable systems that contains both cases I mentioned above?

This is not electromagnetism, since that Hamiltonian looks rather different (I will try to look up the source for that). What I have written down is just the electrostatic potential.

The other strange part is the second Hamiltonian is a real part of the derivative $\frac{d}{dz}$ of my first example.

-