Partial Differential system in two variables with $W(D_6^{(1)})$-symmetry and the Garnier system in two variables

TL;DR

This paper compares the Garnier system in two variables with a four-dimensional PDE system exhibiting $W(D_6^{(1)})$-symmetry, highlighting their differences in compactification but similarities in holomorphy conditions.

Contribution

It provides a detailed comparison between the Garnier system and a related PDE system with affine Weyl group symmetry, revealing structural similarities and differences.

Findings

Both systems share five holomorphy conditions in variables p1, p2.

They differ in their compactification in variables q1, q2.

The systems are connected through their symmetry and holomorphy properties.

Abstract

In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however, has same five holomorphy conditions in the variables $p_1,p_2$.