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In the previous paper hep-th/0604112 we calculated the first of the fiveplanar two-loop diagrams for the Lcc vertex of the general non-AbelianYang-Mills theory, the vertex which allows us in principle to obtain all othervertices via the Slavnov-Taylor identity. The integrand of this first diagramhas a simple Lorentz structure. In this letter we present the result for thesecond diagram, whose integrand has a complicated Lorentz structure. Thecalculation is performed in the D-dimensional Euclidean position space. Weinitially perform one of the two integrations in the position space and thenreduce the Lorentz structure to D-dimensional scalar single integrals. Some ofthe latter are then calculated by the uniqueness method, others by theGegenbauer polynomial technique. The result is independent of the ultravioletand the infrared scale. It is expressed in terms of the squares of spacetimeintervals between points of the effective fields in the position space -- itincludes simple powers of these intervals, as well as logarithms andpolylogarithms thereof, with some of the latter appearing within the Davydychevintegral J(1,1,1). Concerning the rest of diagrams, we present the result forthe contributions correponding to third, fourth and fifth diagrams withoutgiving the details of calculation. The full result for the Lcc correlator ofthe effective action at the planar two-loop level is written explicitly formaximally supersymmetric Yang-Mills theory.Comment: 29 pages, 1 figure, minor changes; three references added, one new  paragraph in Introduction added, Note Added is extended; to appear in JHE

Further results for the two-loopLccvertex in the Landau gauge