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We find and study a six (resp. seven, eight)-parameter family of polynomialHamiltonian systems of second order, respectively. This system admits theaffine 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 firstexample which gave second-order polynomial Hamiltonian system with ${\tildeW}(E_6^{(1)})$ (resp. ${\tilde W}(E_7^{(1)}), W(E_8^{(1)})$)-symmetry.  We also show that its space of initial conditions $S$ is obtained by gluingeight (resp. nine, ten) copies of ${\Bbb C}^2$ via the birational andsymplectic transformations.

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