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

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\mathrm{.}${\tilde W}(E_7^{(1)}), W(E_8^{(1)})$)-symmetry. We also show that its space of initial conditions $S$ is obtained by gluing eight (resp. nine, ten) copies of ${\mathbb{C}}^2$ via the birational and symplectic transformations.