We study local integrability of a plane autonomous polynomial system of ODEs depending on five parameters with a degenerate singular point at the origin. The approach is based on making use of the Power Geometry Method and the computation of normal forms. We look for the complete set of necessary conditions on parameters of the system under which the system is locally integrable near the degenerate stationary point. We found earlier that the sets of parameters satisfying these conditions consist of four two-parameter subsets in the full five-parameter co-space. Now we consider the special subcase of the case b^2 = 2/3 and separate subsubcases when additional first integrals can exist. Here we have found two such integrals. © 2017, Springer International Publishing AG.