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While a local theory of Bonnet surfaces exists for decades, a global theory was an open issue for a long time. In particular the existence of Bonnet surfaces with critical points of the mean curvature function H (in particular umbilic points)
d H = 0
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was not clear.
This is due to the fact that the problem always leads to a highly non-linear differential equation for H. As it comes out this equation processes the Painleve property,
which finally helps in solving the global theory issue.
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