  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 $P_{VI}$.
