We give a detailed study of finite-energy solutions to elliptic sine-Gordon (SG) equation in the plane with point-like singularities. These bound-state solutions (in a sense of scalar field theory) with only one singularity at the origin demonstrate a target-like annular soliton pattern at large distance from the origin. An effective radius of this pattern is calculated both analytically and numerically for the case of axial symmetric solutions. The analytic study is based on an isomonodromic deformation method for the third Painlevé equation, which distinguishes bound-state solutions as separatrices in a manifold of general (infinite-energy) solutions. Exact analytic estimates give us a tool to study bounded-state solutions to the non-integrable SG equation with forcing. Namely, for large intensity at the singularity we derive a critical value of forcing, which governs the existence and stability of the bound-state solutions. This plays a crucial role for two concrete physical applications dealing with large area Josephson junctions and nematic liquid crystals in a rotating magnetic field. For both examples we compute critical values of field and driving forces which enables the formation of modes with finite energy. These numerical computed critical values correlate well with computer simulations and experimental data.