  We use the dynamical mean-field approximation to study singularities in the
self-energy and a two-particle irreducible vertex induced by the
metal-insulator transition of the disordered Falicov-Kimball model. We set
general conditions for the existence of a critical metal-insulator transition
caused by a divergence of the imaginary part of the self-energy. We calculate
explicitly the critical behavior of the self-energy for the symmetric and
asymmetric disorder distributions. We demonstrate that the metal-insulator
transition is preceded by a pole in a two-particle irreducible vertex. We show
that unlike the singularity in the self-energy the divergence in the
irreducible vertex does not lead to non-analyticities in measurable physical
quantities. We reveal universal features of the critical metal-insulator
transition that are transferable also to the Mott-Hubbard transition in the
models of the local Fermi liquid.
