On some Hamiltonian structures of coupled Painlev\'e II systems in dimension four

We find and study a two-parameter family of coupled Painlev'e II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables $t,s$. Setting $s=0$, we can obtain an autonomous version of the coupled Painlev'e II systems. We also show its symmetry and holomorphy conditions.