  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 ${\rm SU}_2$ on $S^4$, but which are not globally defined. We will
see that these instantons produce solutions to a one parameter family of
Painlev\'e VI equations ($\text{P}_{\text{VI}}$) and we will give an explicit
expression of the map between instantons and solutions to
$\text{P}_{\text{VI}}$. 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 [Mu\~niz Manasliski R., Contemp. Math.,
Vol. 434, Amer. Math. Soc., Providence, RI, 2007, 215-222] and [Mu\~niz
Manasliski R., J. Geom. Phys. 59 (2009), 1036-1047, arXiv:1602.07221], where
instantons without singularities are studied.
