Analogue of the universal law for general elliptic KdV potentials

Ascertain whether the differential relation dτ ∧ dB ≡ 8π dr ∧ ds holds for the monodromy map (τ, B) → (r, s) associated with second-order equations y''(z) = [∑_{j=1}^n m_j(m_j + 1)℘(z − p_j(τ); τ) + B] y(z), where q(z; τ) = ∑_{j=1}^n m_j(m_j + 1)℘(z − p_j(τ); τ) is an elliptic KdV potential satisfying the constraints (1.16) and Q_p(B; τ) denotes the corresponding spectral polynomial, thereby extending the universal law proved for Darboux–Treibich–Verdier potentials.

Background

The paper proves a universal law for Darboux–Treibich–Verdier (DTV) potentials: the map from (τ, B) to monodromy data (r, s) is holomorphic, locally one-to-one, and satisfies dτ ∧ dB ≡ 8π dr ∧ ds. The authors then consider more general elliptic KdV potentials, characterized by Gesztesy–Unterkofler–Weikard, of the form q(z; τ) = ∑ m_j(m_j + 1)℘(z − p_j(τ); τ) up to a constant, with positions p_j(τ) subject to the constraints in (1.16).

For these general potentials, the associated second-order equation y''(z) = q(z; τ) + B has a spectral polynomial Q_p(B; τ), and a monodromy map (τ, B) ↦ (r, s) analogous to the DTV case is well-defined on {(τ, B) | Q_p(B; τ) = 0}. The authors ask whether the same universal law relating parameter differentials and monodromy coordinates extends to this broader class.