Use of the inverse problem method for solving certain classical problems of differential geometry

By using the dressing method, which is a modern version of the inverse problem method, it is possible to solve the following classical problems of differential geometry.
1. Classification of $N$-orthogonal curvilinear coordinate systems in Euclidean space and in a space of constant curvature. (Formulated in 1813 by Binnet and Dupin.)
2. Description of spaces of flat connections together with their embedding in a Euclidean space.