A shortcut to the Kovalevskaya curve

TL;DR

This paper introduces a systematic method to derive algebraic separation curves for the Kovalevskaya top and its generalizations, connecting spectral curves with Prym varieties, and enabling the discovery of new separation curves.

Contribution

It provides a new algorithmic approach to obtain separation curves from spectral data, applicable to general constants of motion and generalizations of the Kovalevskaya top.

Findings

Derived the Kovalevskaya separation curve from spectral data

Developed an algorithm for general constants of motion

Discovered new separation curves for generalized systems

Abstract

We present a systematic way of derivation of the algebraic curves of separation of variables for the classical Kovalevskaya top and its generalizations, starting from the spectral curve of the corresponding Lax representation found by Reyman and Semonov-Tian-Shansky. In particular, we show how the known Kovalevskaya curve of separation can be obtained, by a simple one-step transformation, from the spectral curve. The algorithm works for the general constants of motion of the system and is based on W. Barth's description of Prym varieties via pencils of genus 3 curves. It also allows us to derive new curves of separation of variables in various generalizations of this system.