Abstract
We build analytic surfaces in R3(c) represented by the most general sixth Painlevé equation PVI in two steps. First, the moving frame of the surfaces built by Bonnet in 1867 is extrapolated to a new, second order, isomonodromic matrix Lax pair of PVI, whose elements depend rationally on the dependent variable and quadratically on the monodromy exponents θj. Second, 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 PVI.
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