Systems of Kowalevski type, discriminantly separable polynomials and quad-graphs

TL;DR

This paper introduces a new class of integrable systems linked to discriminantly separable polynomials, providing explicit solutions, classification, and connections to geometric structures like conic pencils and quad-graphs.

Contribution

It defines and classifies discriminantly separable polynomials of degree two, constructs new integrable systems of Kowalevski type, and relates these to geometric structures and discrete quad-graph systems.

Findings

Constructed explicit genus two theta-function solutions.

Classified discriminantly separable polynomials via zero structures.

Linked polynomial classification to pencils of conics and quad-graphs.

Abstract

We establish a new class of integrable {\it systems of Kowalevski type}, associated with discriminantly separable polynomials of degree two in each of three variables. Defining property of such polynomials, that all discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each (denote one of the polynomial components as $P$), lead to an effective integration procedure. In the motivating example, the celebrated Kowalevski top, the discriminant separability is a property of the polynomial defining the Kowalevski fundamental equation. We construct several new examples of systems of Kowalevski type, and we perform their explicit integration in genus two theta-functions. One of the main tasks of the paper is to classify such discriminantly separable polynomials. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. From a geometric point of view, such a classification is related to the types of pencils of conics. We construct also discrete integrable systems on quad-graphs associated with discriminantly separable polynomials. We establish a relationship between our classification and the classification of integrable quad-graphs which has been suggested recently by Adler, Bobenko and Suris. As a fit back, we get a geometric interpretation of their results in terms of pencils of conics, and in the case of general position, when all four zeros of the polynomial $P$ are distinct, we get a connection with the Buchstaber-Novikov two-valued groups on $\mathbb {CP}^1$.