  We investigate exceptional points, which are branch point singularities of
two resonance eigenstates, in spectra of the hydrogen atom in crossed external
electric and magnetic fields. A procedure to systematically search for
exceptional points is presented, and their existence is proven. The properties
of the branch point singularities are discussed with effective low-dimensional
matrix models, their relation with avoided level crossings is analyzed, and
their influence on dipole matrix elements and the photoionization cross section
is investigated. Furthermore, the rare case of a connection between three
resonances almost forming a triple-degeneracy in the form of a cubic root
branch point is discussed.
