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