Painlevé property for the full 3D system (1.1)

Determine whether the non-autonomous three-dimensional system (1.1), without restricting solutions to the invariant hypersurface St defined by h(x,y,z;t)=0, possesses the Painlevé property (i.e., that all movable singularities of its solutions are poles).

Background

The paper studies a constrained 3D system of first-order ODEs (equation (1.1)) arising from semi-classical orthogonal polynomials, and shows that its restriction to a parametrized invariant hypersurface can be reduced to two distinct 2D systems identified with Painlevé VI, establishing the Painlevé property in those restricted cases.

The authors further analyze an autonomous limit of the 3D system and show it is Liouville–Poisson integrable with elliptic-fibration level sets, which suggests—but does not prove—that the full, unrestricted 3D system may inherit the Painlevé property. Establishing this directly for the full non-autonomous 3D system remains unresolved.