A new hierarchy of integrable systems associated to Hurwitz spaces

TL;DR

This paper introduces a new class of integrable systems linked to Hurwitz spaces, generalizing the Ernst equation and providing a geometric framework connecting meromorphic functions, flat metrics, and hydrodynamic systems.

Contribution

It presents a novel hierarchy of integrable systems associated with Hurwitz spaces, extending existing deformation schemes and linking solutions to flat diagonal metrics.

Findings

New integrable systems associated with Hurwitz spaces introduced.

Solutions define flat Darboux-Egoroff metrics and hydrodynamic systems.

Generalizes the Ernst equation and deformation schemes.

Abstract

In this paper we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of "times". Our systems give a natural generalization of the Ernst equation; in genus zero they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux-Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.