On binary differential equations

In this paper we give the local classification of solution curves of binary differential equations a(x,y)dy 2 +2b(x,y)dxdy+c(x,y)dx 2 =0 at points at which the discriminant function b 2 -ac has a Morse singularity. We also discuss the formal reduction of such equations to some normal form. The results determine the topological structure of asymptotic curves on a smooth surface with a flat umbilic, the principal curves at general umbilics, and asymptotic curves at cross-cap points of an otherwise smooth surface.