Generalized Bonnet surfaces and Lax pairs of ${\mathrm{P}}_{\mathrm{V}\mathrm{I}}$

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 $P_{VI}$.