Ordinary differential systems in dimension three with affine Weyl group symmetry of types $D_4^{(1)},B_3^{(1)},G_2^{(1)},D_3^{(2)}$ and $A_2^{(2)}$

TL;DR

This paper introduces a family of three-dimensional differential systems with affine Weyl group symmetries of various types, reduces them to Painlevé equations, and explores their symmetries, invariants, and parameter restrictions.

Contribution

It presents new systems with specific affine Weyl group symmetries and analyzes their properties, including invariants and reductions, which were not previously known.

Findings

New four-parameter family of systems with $D_4^{(1)}$ symmetry.

Reduction to Painlevé type equations via first integrals.

Identification of symmetries and invariant divisors for each system.

Abstract

We present a four-parameter family of ordinary differential systems in dimension three with affine Weyl group symmetry of type $D_4^{(1)}$. By obtaining its first integral, we can reduce this system to the second-order non-linear ordinary differential equations of Painlev\'e type. We also study this system restricted its parameters. Each system can be obtained by connecting some invariant divisors in the system of type $D_4^{(1)}$. Each system admits affine Weyl group symmetry of types $B_3^{(1)},G_2^{(1)},D_3^{(2)}$ and $A_2^{(2)}$, respectively. These symmetries, holomorphy conditions and invariant divisors are new.