Reply to "Comment" by A. V. Tsiganov

TL;DR

This paper derives explicit Abel-Jacobi equations for the Goryachev case using separation variables on a hyperboloid, highlighting their non-commuting properties similar to Kowalevski's shifted variables.

Contribution

It provides explicit forms of Abel-Jacobi equations with degree-six polynomials for the Goryachev case, using specific non-commuting separation variables.

Findings

Explicit Abel-Jacobi equations with degree-six polynomial

Separation variables chosen from hyperboloid parameters

Non-commuting nature of the variables

Abstract

For the Goryachev case we obtain, in the explicit form, the Abel-Jacobi equations with the polynomial of degree six under the radical. We choose the parameters of two families of linear generators of a one sheet hyperboloid to be the separation variables. These variables, as well as the shifted separation variables in the original work of S. Kowalevski, do not commute.