Painlevé property conjecture for the full constrained 3D system

Consider the full constrained 3D system studied in the paper, rather than either of its restrictions to an invariant hypersurface. The Painlevé property means that its solutions have the Painlevé property in the geometric sense used for spaces of initial conditions. Painlevé property conjecture. The 3D system possesses the Painlevé property, and it is possible to construct a space of initial conditions for it whose fibres are rational threefolds with zero-dimensional anti-canonical linear system.

The conjecture is motivated by the authors' proof that an autonomous limit of the system is multi-Hamiltonian and defines an elliptic fibration, together with the known relationship between elliptic fibrations and Painlevé equations. The full 3D system is not established to have the Painlevé property, so the conjecture remains open.