Links from second-order Fuchsian equations to first-order linear systems

The number of parameters in the linear Fuchsian system with four singularities is larger than that in the second-order Fuchsian equation with the same singularities. Hence, in order to find a relation between the given system and the equation it is needed to simplify the matrices – residues at finite Fuchsian singularities. The way to do it is studied. Such approach gives also the possibility to find the relation between the use of the antiquantization procedure and the isomonodromic property for derivation the Painlevé equation $P^{VI}$.