A novel reduction of rational and cubic Hamiltonian systems to Painleve equations

In this paper, we focus on rational and cubic Hamiltonian systems closely related to the Painleve equations. We prove that, after applying an iterated polynomial regularisation procedure, these systems become birationally equivalent to the standard Okamoto polynomial Hamiltonian systems, thus providing an explicit identification with the classical forms of the Painleve III – VI equations. One of the additional elements in our approach is the systematic analysis of the Newton polygons for the polynomial Hamiltonians, including how their area and shape change under each iteration of the regularisation procedure. This analysis helps identify the correspondence with the standard Okamoto polynomial Hamiltonians for the Painleve equations.