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@@ -1,1 +1,1 @@ - +An internal condition for figuring out special values of the accessory parameters is given. Some observations on the coefficients of this local expansion are explained using a result of Takhtajan and Zograf and some special symmetry. We give a simplified proof of the virtually everywhere real analyticity within the case of a single accessory parameter as it occurs e.g. within the sphere topology with four sources or for the torus topology with a single source through the use of only the general analyticity properties of the solution of the auxiliary equation. We are going to prove a extra general end result, of which the above theorem is a corollary. On this section we'll present that, beneath sure assumptions, the converse is also true. We present that underneath an irreducibility assumption one reaches the final result of the a.e. Otherwise one can lengthen the area of convergence in the following manner. 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