Abstract
In Nabarro (2011), we define and study the families of oe"-conjugate curve congruences and the families of reflected oe"-conjugate curve congruences , i = 1, 2, associated to a self-adjoint operator oe" on a smooth and oriented surface M endowed with a Lorentzian metric. These families parametrize parts of the pencils of forms that link the equation of the oe"-asymptotic (resp. oe"-characteristic) curves and that of the oe"-principal curves. There is a crucial difference with the Riemannian case due to the existence of lightlike curves. In this paper, we study the generic local singularities in the members of these families and describe the way they bifurcate within the families.