Conjecture on Painlevé property and space of initial conditions for the full 3D system (1.1)

Prove that the non-autonomous three-dimensional system (1.1) possesses the Painlevé property, and construct a space of initial conditions for it whose fibres are rational threefolds with zero-dimensional anti-canonical linear system.

Background

The authors’ geometric and Hamiltonian analysis of two different restrictions of the 3D system to an invariant hypersurface shows these 2D systems have the Painlevé property and are birationally equivalent to Painlevé VI. They also establish that an autonomous limit of the 3D system is Liouville–Poisson integrable with elliptic curves as level sets.

Based on these results and known correspondences between elliptic fibrations and Painlevé equations, the authors conjecture that the full 3D system shares the Painlevé property and admits a higher-dimensional space of initial conditions analogous to those in the Okamoto–Sakai theory.