Appearance of Z_{r,s}^{(n)}(τ) in denominators of Painlevé VI solutions

Demonstrate that for Darboux–Treibich–Verdier potentials, the pre-modular forms Z_{r,s}^{(n)}(τ) appear in the denominators of the expressions of solutions to Painlevé VI in its elliptic form, generalizing the established behavior for Z_{r,s}^{(n,0,0)}(τ) in the Lamé case.

Background

In prior work, the authors proved the simple-zero property of Z_{r,s}^{(n,0,0)}(τ) by showing that these pre-modular forms occur in denominators of certain Painlevé VI solutions, linking zero multiplicities to analytic behavior of the solutions.

In this paper they extend the simple-zero property to the DTV case and develop deeper connections between Painlevé VI and pre-modular forms, but they explicitly state that confirming the analogous denominator appearance for Z_{r,s}^{(n)}(τ) in the more general setting has not yet been achieved.