Moduli space of rank three logarithmic connections on the projective line with three poles

In this paper, we describe the moduli space of rank three parabolic logarithmic connections on the projective line with three poles for any local exponents. In particular, we show that the family of moduli spaces of rank three parabolic $\phi$-connections on the projective line with three poles is isomorphic to the family of $A^{(1)*}_2$-surfaces in Sakai's classification of Painlev\'e equations. Through this description, we investigate the relation between the apparent singularities and underlying parabolic bundles.