Geometric regularisation of a Hamiltonian system related to PIV via Painlev´e-Calogero correspondence

We consider a Hamiltonian system obtained by Takasaki, related by an algebraic transformation to the fourth Painlev´e equation PIV under the so-called Painlev´e-Calogero correspondence. The Hamiltonian in question is associated to the rank-one case of an extended Calogero system with rational potential. The algebraic transformation gives a map from this system to the Okamoto Hamiltonian form of PIV, which possesses the Painlev´e property in the sense that all solutions are single-valued about movable singularities. This property is closely related to the existence of a space of initial conditions for the system, first constructed by Okamoto, which is a bundle of open rational surfaces of which the flow of the system defines a uniform foliation. However, the algebraic map to the Okamoto system is not birational and the Takasaki Hamiltonian system does not enjoy the Painlev´e property, so does not possess a space of initial conditions in the sense of Okamoto.

We show that it is still possible to associate a bundle of rational surfaces to the Takasaki Hamiltonian system on which it admits certain regularising transformations. This detects and recovers the algebraic transformation to the Okamoto system with the Painlev´e property, and we discuss how this realises the surface from the Okamoto Hamiltonian as a quotient of that from the Takasaki system.

Based on joint work with Galina Filipuk