The sine-Gordon equation is a nonlinear partial differential equation that finds a wealth of applications. One method for finding analytical solutions is via an ansatz including Jacobi elliptic functions. The aim of this paper is to analyse the special cases of these solutions of the \((2+1)\) -dimensional sine-Gordon equation, where the elliptic modules assume their limiting values. These cases are collected, regions of existence are provided and connections to lower dimensional sine-Gordon equations are discussed. The results are applied in the context of two-dimensional Josephson junctions. Electric and magnetic fields of such objects are investigated for different combinations of solutions and their respective allowed parameters.