Hamiltonians, Their Newton Polygons and Painlevé Equations

We study selected cases of the Hamiltonian of bi-degree (3, 2) with arbitrary analytic coefficients. By imposing constraints on the coefficient functions, which can lower the bi-degree, we can recover the well-known polynomial Hamiltonians with rational coefficients for the Painlevé equations in the regularisation of the corresponding Hamiltonian system. Moreover, for the first and the second Painlevé equations we can also recover polynomial Hamiltonians with meromorphic coefficients. In addition, we examine various shapes of the Newton polygons and graphically visualize the coalescence of the Painlevé equations via inclusion of the Newton polygons into each other.