Bonnet Surfaces in 3-Space
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
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.