Space of initial conditions for the full 3D system (1.1)

Ascertain whether it is possible to construct a space of initial conditions, in the sense of Definition 2.12, for the non-autonomous three-dimensional system (1.1) itself (i.e., without restricting to the invariant hypersurface), thereby providing a compactified phase space with a uniform foliation by solution leaves.

Background

While the authors build spaces of initial conditions for the two restricted 2D systems and relate them to the standard Painlevé VI geometry, they emphasize that extending such a construction to the full 3D system is substantially harder due to higher-dimensional geometric complexities and the presence of different types of indeterminacies.

They indicate that the autonomous limit exhibits properties (e.g., elliptic fibration, multi-Hamiltonian structure) suggestive of Painlevé-type behavior, motivating the question of whether an appropriate higher-dimensional analogue of the Okamoto–Sakai construction exists for the full 3D system.