Relationship between the rank-two and rank-three Riemann–Hilbert maps for Painlevé IV

Determine the relationship between the Riemann–Hilbert map arising from the rank-two linear problem for Painlevé IV and the Riemann–Hilbert map arising from the rank-three linear problem, including whether the two constructions are compatible.

Background

The paper establishes the symmetry result for Painlevé IV using a rank-two linear problem. A different rank-three linear problem is known in the literature and has its own Riemann–Hilbert map to a corresponding monodromy surface.

Although the paper describes a commutative diagram that could provide an alternative route to the symmetry theorem, it does not determine the induced map between the two monodromy-surface models or otherwise resolve the relationship between the rank-two and rank-three Riemann–Hilbert constructions.

References

There is another Riemann-Hilbert map coming from a rank 3 linear problem associated with $P{IV}$, and the relation between these is unknown, see \Cref{rem:commutingRHsPIV}.

— On monodromy of monodromy surfaces  (2608.25594 - Roffelsen et al., 26 Aug 2026) in Section 1, Introduction, paragraph discussing the Painlevé IV Riemann–Hilbert maps; see also Remark \ref{rem:commutingRHsPIV}