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Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with threesingular points and one accessory parameter are presented. The shift operatorsfor $H_6$ are studied, which lead to assign the accessory parameter of $H_6$ acubic polynomial of local exponents so that the equations have nice properties:shift operators and several symmetries. The other equations will be studied inthe forthcoming papers.

Fuchsian differential equations of order 3,...,6 with three singular points and an accessory parameter I, Equations of order 6