Coupled Hamiltonian systems with extended affine Weyl group symmetry of type $D_3^{(2)}$

TL;DR

This paper introduces a new family of five-dimensional differential systems with affine Weyl group symmetry of type D_3^{(2)}, expanding the class of higher order Painlevé systems and deriving related coupled Hamiltonian systems.

Contribution

It presents the first example of higher order Painlevé type systems with D_3^{(2)} symmetry and derives associated coupled Hamiltonian systems with polynomial Hamiltonians.

Findings

Identified a two-parameter family of 5D differential systems with D_3^{(2)} symmetry.

Established symmetry and holomorphy conditions for the system.

Derived polynomial first integrals leading to 4D coupled Hamiltonian systems.

Abstract

We find a two-parameter family of ordinary differential systems in dimension five with the affine Weyl group symmetry of type $D_3^{(2)}$. We show its symmetry and holomorphy conditions. This is the second example which gave higher order Painlev\'e type systems of type $D_{3}^{(2)}$. By obtaining its first integrals of polynomial type, we can obtain a two-parameter family of coupled Hamiltonian systems in dimension four with the polynomial Hamiltonian.