Middle convolution and addition are operations for Fuchsian systems of differential equations which preserve the number of accessory parameters. In this paper we show that they also preserve the deformation equations. Several Bäcklund transformations of the sixth Painlevé equation are obtained from this viewpoint.

We show that every accessory parameter free system of differential equations of Okubo normal form has integral representation of solutions. The proof is constructive; we study the change of solutions under the operations}the extension and the restriction, which have been introduced by Yokoyama [Construction of systems of differential equations of Okubo normal form with rigid monodromy, preprint] in order to construct every such system of differential equations. Several examples are given. # 2002 Elsevier Science (USA)

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