Rank-1 codimension one singularities of positive quadratic differential forms

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D23-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D2 and the other of type D3) which collapse at the D23-singular point when the bifurcation parameter is reached, and afterwards disappear. In local (x,y)-coordinates, such a point appears at the origin of a planar differential equation of the form a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2, with (b^2-ac)(x,y)>= 0, such that (1) the first jet of the map (a,b,c) at the origin is T1(a,b,c) (0,0)=(y,0,-y) and (2)(delta)^2b/(delta)x^2(not equal)0.