Singular Instantons and Painlevé VI

We consider a two parameter family of instantons, which is studied in [Sadun L., Comm. Math. Phys. 163 (1994), 257–291], invariant under the irreducible action of $\mathrm{SU}_2$ on $S^4$, but which are not globally defined. We will see that these instantons produce solutions to a one parameter family of Painlevé VI equations ($\mathrm{P_{VI}}$) and we will give an explicit expression of the map between instantons and solutions to $\mathrm{P_{VI}}$. The solutions are algebraic only for that values of the parameters which correspond to the instantons that can be extended to all of $S^4$. This work is a generalization of [Muñiz Manasliski R., Contemp. Math., Vol. 434, Amer. Math. Soc., Providence, RI, 2007, 215–222] and [Muñiz Manasliski R., J. Geom. Phys. 59 (2009), 1036–1047, arXiv:1602.07221], where instantons without singularities are studied.