Only with 1-D Green function for 1-D elliptic differential operator, we can solve 2-D/3-D general elliptic problems by applying the axial Green function method (AGM). An extension of AGM is proposed to enforce Neumann boundary conditions. This extension is directly available for 2-D problems with straight boundaries parallel to axes on which Neumann boundary conditions are assigned. It is thoroughly attributed to the specific axial Green functions associated with the Neumann conditions. Moreover, since this extended AGM (XAGM) in 1-D satisfies the transmission condition across an interface along which the permittivity is discontinuous, it can be applied to 2-D problems with interfaces parallel to axes without loss of accuracy. Finally, we apply the XAGM in 2-D to 3-D axisymmetric electric potential problems with variable and/or even discontinuous permittivities along interfaces. Owing to the cylindrical coordinate transform, the transformed problem is tractable to solve using this XAGM. Arbitrary distribution of axial lines is available, which must be a marked advantage of XAGM compared to other methods.