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 D-23-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D-2 and the other of type D-3) which collapse at the D-23-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) dx dy + c(x, y) dx(2), with (b(2) - ac)(x, y)greater than or equal to 0, such that (1) the first jet of the map (a, b, c) at the origin is T-1 (a, b, c)(0, 0) = (y, 0, -y) and (2) partial derivative(2)b/partial derivativex(2) not equal 0 (C) 2004 Elsevier Inc. All rights reserved.