Designing and constructing underground structures for various purposes, such as tunnels, mines, mine workings, necessitate the development of procedures for calculating their strength and reliability. The physical model of such objects worth considering is a homogeneous isotropic half-space that contains an infinitely long hollow cylinder, located parallel to its border. One can explore problems related to the mechanics of deformable solids for such a multiply connected body. This paper reports the proofs of addition theorems for the basic solutions to the Lamé equation regarding the half-space and cylinder recorded, respectively, in the Cartesian and cylindrical coordinate systems. This result is important from a theoretical point of view in order to substantiate a numerical-analytical method ‒ the generalized Fourier method. This method makes it possible to solve spatial boundary problems from the theory of elasticity and thermo-elasticity for isotropic and transversal-isotropic multiply connected bodies. Similar to the classical Fourier method, the general solutions to equilibrium equations have been used here but in several coordinate systems rather than one. From a practical point of view, this method has made it possible to investigate the combined problem of elasticity theory regarding the multiply-connected body described above. The analysis of the stressed-strained state of this elastic body has made it possible to draw conclusions on determining those regions that are most vulnerable to destruction. The highest values are accepted by normal stresses in the region between the boundaries of the half-space and the cylinder. Changing the σy component along the Ox axis corresponds to the displacements assigned on the half-space. The τxy component contributes less to the distribution of stresses than σx and σy. The applied aspect of using the reported results is the possibility to apply them when designing underground structures