Level sets of functions and symmetry sets of smooth surface sections
  We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points.   We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4].   However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness. 
Andre Diatta, Peter Giblin, Brendan Guilfoyle and Wilhelm Klingenberg