Generalized Bonnet surfaces and Lax pairs of ${{\rm P_{\rm VI}}}$

TL;DR

This paper extends classical Bonnet surfaces to a broader class associated with the sixth Painlevé equation by constructing a new Lax pair and deriving a quantum correspondence, enriching the geometric and analytical understanding of $P_{VI}$.

Contribution

It introduces a novel Lax pair for $P_{VI}$ dependent on monodromy exponents and extends Bonnet surfaces to include additional degrees of freedom, linking geometry with integrable systems.

Findings

Constructed a second order Lax pair for $P_{VI}$ with rational dependence on variables.

Extended Bonnet surfaces to include two extra degrees of freedom.

Derived a rigorous quantum correspondence for $P_{VI}$.

Abstract

We build analytic surfaces in $\mathbb{R}cubec$ represented by the most general sixth Painlev\'e equation $P_{VI}$ in two steps. Firstly, the moving frame of the surfaces built by Bonnet in 1867 is extrapolated to a new, second order, isomonodromic matrix Lax pair of $P_{VI}$, whose elements depend rationally on the dependent variable and quadratically on the monodromy exponents $\theta_j$. Secondly, by converting back this Lax pair to a moving frame, we obtain an extrapolation of Bonnet surfaces to surfaces with two more degrees of freedom. Finally, we give a rigorous derivation of the quantum correspondence for $P_{VI}$.