Symmetric Hamiltonian of the Garnier system and its degenerate systems in two variables

TL;DR

This paper introduces symmetric Hamiltonians for degenerate Garnier systems in two variables, explores their symmetry and holomorphy conditions, and generalizes these systems through an inductive approach, illustrating their confluence with Painlevé systems.

Contribution

It provides the first explicit symmetric Hamiltonians for degenerate Garnier systems and extends these systems via an inductive framework incorporating symmetry and holomorphy.

Findings

Symmetric Hamiltonians for degenerate Garnier systems are constructed.

The systems are generalized through an inductive process.

Confluence among systems is demonstrated via Painlevé confluence.

Abstract

We present {\it symmetric Hamiltonians} for the degenerate Garnier systems in two variables. For these symmetric Hamiltonians, we make the symmetry and holomorphy conditions, and we also make a generalization of these systems involving symmetry and holomorphy conditions inductively. We also show the confluence process among each system by taking the coupling confluence process of the Painlev\'e systems.