Unfolding of eigenvalue surfaces near a diabolic point due to a complex   perturbation
  The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied. 
O.N. Kirillov, A.A. Mailybaev, and A.P. Seyranian