Explicit Hamiltonian structure of the Flaschka-Newell Painlev\'{e} II hierarchy via symmetry reduction of the Painlev\'{e} IV hierarchy

We study the Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the associated space of meromorphic connections.

Building on the realization of this hierarchy as a reduction of the Painlevé IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution.

After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold.

This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlevé II hierarchy.

This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.