From Heun Class Equations to Painlevé Equations

In this paper, we aim at addressing the globalization problem of Hamilton–DeDonder–Weyl equations on a local k-symplectic framework and we introduce the notion of locally conformal k-symplectic (l.c.k-s.) manifolds. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton–Jacobi equation, which is specifically proposed in a l.c.k-s manifold.