Connection between complete and Möbius forms of gauge invariant operators

We study the connection between complete representations of gauge invariant operators and their Möbius representations acting in a limited space of functions. The possibility to restore the complete representations from Möbius forms in the coordinate space is proven and a method of restoration is worked out. The operators for transition from the standard BFKL kernel to the quasi-conformal one are found both in Möbius and total representations.