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

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