  The canonical phase diagram of the Blume-Emery-Griffiths (BEG) model with
infinite-range interactions is known to exhibit a fourth order critical point
at some negative value of the bi-quadratic interaction $K<0$. Here we study the
microcanonical phase diagram of this model for $K<0$, extending previous
studies which were restricted to positive $K$. A fourth order critical point is
found to exist at coupling parameters which are different from those of the
canonical ensemble. The microcanonical phase diagram of the model close to the
fourth order critical point is studied in detail revealing some distinct
features from the canonical counterpart.
