After determining the relationship between cylindrical and spherical solid vector harmonics, we arrive at the complete and effective solution for the static deformation due to a multipolar source of arbitrary order in a homogeneous isotropic half-space. Using the correpsondence between spherical harmonics and seismic moment tensors obtained earlier, we construct a catalog of mirror image sources which yield zero traction on the plane. For each angular order number l the total number of these images is 15 for a general case, of which the first spheroidal source has seven images, the toroidal source has four images, and the second spheroidal source also has four images. We derive the conditions under which image point sources are converted into semi-infinite line nuclei. We analyze several special cases of deformation in a half-space, namely, axisymmetric, generalized Boussinesq-Cerruti, thermoelastic deformations. We present several general and specific models of finite sources of elastic displacement in a half-space and discuss possibilities for inverting complex extended sources of elastic deformation. The nonuniqueness of the inversion can be estimated objectively.