  We study a one-parameter family of the fourth-order ordinary differential
equations obtained by similarity reduction of the modifed Sawada-Kotera
equation. We show that the birational transformations take this equation to the
polynomial Hamiltonian system in dimension four. We make this polynomial
Hamiltonian from the viewpoint of accessible singularity and local index. We
also give its symmetry and holomorphy conditions. These properties are new.
Moreover, we introduce a symmetric form in dimension five for this Hamiltonian
system by taking the two invariant divisors as the dependent variables. Thanks
to the symmetric form, we show that this system admits the affine Weyl group
symmetry of type $A_2^{(2)}$ as the group of its B{\"a}cklund transformations.
