L. Losonczi  determined local solutions of the generalized Cauchy equation f(F(x, y))= f(x) + f(y) on components of the denition of a given associative rational function F. The class of the associative rational function was described by A. Chéritat  and his work was followed by paper  of the author. The aim of the present paper is to describe local solutions of the equation considered for some singular associative rational functions.
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