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We find a one-parameter family of polynomial Hamiltonian system in twovariables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can beobtained by the compatibility conditions for the linear differential equationsin three variables. We give a relation between it and the second member of thesecond Painlev\'e hierarchy. Moreover, we give some relations between anautonomous version of its polynomial Hamiltonian system in two variables andthe mKdV hierarchies.

Polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry and the second Painlev\'e hierarchy