(joint work with M. Vanlessen) We consider the following fourth order analogue of the Painlev\'e I equation, \[ x=Ty-\left(\frac{1}{6}y^3+\frac{1}{24}(y_x^2+2yy_{xx}) +\frac{1}{240}y_{xxxx}\right). \] We give an overview of how to prove the existence of a real solution $y$ with no poles on the real line, which was conjectured by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for $y(x,T)$ as $x\to\pm\infty$. Furthermore, we explain how functions associated with the $P_I^2$ equation appear in a double scaling limit near singular edge points in random matrix models.

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