Painlevé scheme

الأصل · EN

In this note, we review the notion of Painlevé scheme of the sixth Painlevé equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two Σ₂. The key method is Painlevé α-method for each accessible singular point. Giving a Painlevé scheme in the differential system satisfying certain conditions, we can recover the Painlevé VI system with the polynomial Hamiltonian. We also consider the case of the Painlevé V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular (n+2)-points in the Hirzebruch surface of degree n; Σₙ. This equation has symmetry of symmetric group of degree n+2.