  Motivated by the pseudogap state of the cuprates, we introduce the concept of
an "exceptional" van Hove singularity that appears when strong
electron-electron interaction splits an otherwise simply connected Fermi
surface into multiply connected pieces. The singularity describes the touching
of two pieces of the split Fermi surface. We show that this singularity is
proximate to a second order van Hove singularity, which can be accessed by
tuning a dispersion parameter. We argue that, in a wide class of cuprates, the
end-point of the pseudogap is accessed only by triggering the exceptional van
Hove singularity. The resulting Lifshitz transition is characterized by
enhanced specific heat and nematic susceptibility, as seen in experiments.
