Realization of symmetry of $A_2^{(1)\ast }$-surfaces as transformations of logarithmic connections

An $A_2^{(1)\ast }$-surface is a space of initial conditions of certain difference Painlevé equations. $A_2^{(1)\ast }$-surfaces are realized as the moduli spaces of parabolic logarithmic connections. In this paper, we realize the symmetry of $A_2^{(1)\ast }$-surfaces as transformations of parabolic logarithmic connections.