Reduction of divisors and Clebsch system

TL;DR

This paper investigates the properties of reduced divisors on spectral curves associated with the Clebsch system's Lax matrices, extending the analysis to higher-dimensional cases using algebraic geometry principles.

Contribution

It introduces a study of reduced divisors on spectral curves for the Clebsch system and generalizes the results to $gl^*(n)$ cases, linking integrable systems with algebraic geometry.

Findings

Characterization of reduced divisors on spectral curves

Extension of divisor reduction to $gl^*(n)$ systems

Insights into the algebraic structure of Lax matrices

Abstract

There are a few Lax matrices of the Clebsch system. Poles of the Baker-Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann-Roch theorem, each class has a unique reduced representative. We discuss properties of such reduced divisor on the spectral curve of $3\times 3$ Lax matrix having a natural generalization to $gl^*(n)$ case.