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We build analytic surfaces in $\mathbb{R}cubec$ represented by the mostgeneral sixth Painlev\'e equation $P_{VI}$ in two steps. Firstly, the movingframe of the surfaces built by Bonnet in 1867 is extrapolated to a new, secondorder, isomonodromic matrix Lax pair of $P_{VI}$, whose elements dependrationally on the dependent variable and quadratically on the monodromyexponents $\theta_j$. Secondly, by converting back this Lax pair to a movingframe, we obtain an extrapolation of Bonnet surfaces to surfaces with two moredegrees of freedom. Finally, we give a rigorous derivation of the quantumcorrespondence for $P_{VI}$.

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