Second order polynomial Hamiltonian systems with ${\tilde W}(E_6^{(1)}),{\tilde W}(E_7^{(1)})$ and $W(E_8^{(1)})$-symmetry

TL;DR

This paper introduces new second-order polynomial Hamiltonian systems with affine Weyl group symmetries of types E6^{(1)}, E7^{(1)}, and E8^{(1)}, providing the first such examples with these symmetries.

Contribution

It constructs and analyzes the first known second-order polynomial Hamiltonian systems exhibiting these specific affine Weyl group symmetries.

Findings

Systems have six, seven, and eight parameters.

Space of initial conditions constructed by gluing multiple copies of a2^2.

First examples with these symmetries for second-order polynomial Hamiltonian systems.

Abstract

We find and study a six (resp. seven, eight)-parameter family of polynomial Hamiltonian systems of second order, respectively. This system admits the affine Weyl group symmetry of type $E_6^{(1)}$ (resp. $E_7^{(1)}, E_8^{(1)}$) as the group of its B{\"a}cklund transformations. Each system is the first example which gave second-order polynomial Hamiltonian system with ${\tilde W}(E_6^{(1)})$ (resp. ${\tilde W}(E_7^{(1)}), W(E_8^{(1)})$)-symmetry. We also show that its space of initial conditions $S$ is obtained by gluing eight (resp. nine, ten) copies of ${\Bbb C}^2$ via the birational and symplectic transformations.