Elliptic Families of Solutions of the Kadomtsev–Petviashvili Equation and the Field Elliptic Calogero–Moser System

We present a Lax pair for the field elliptic Calogero–Moser system and establish a connection between this system and the Kadomtsev–Petviashvili equation. Namely, we consider elliptic families of solutions of the KP equation such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is described by a reduction of the field elliptic CM system.
We construct a wide class of solutions to the field elliptic CM system by showing that any $N$-fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles.