The Earth's gravity field may significantly modify static deformations of large earthquakes. In the elastic-gravitational dislocation theory, which was first published in the early 1980s, the gravity effect is taken into account by including a constant external gravity in plane-Earth models. When using this theory, however, the response functions of the models exhibit singularities below a critical wave number, if the models contain an infinite compressible half space medium. Furthermore, the previous algorithm treated these singularities in an incorrect mathematical way. It is shown here that the boundary value problem for such plane-Earth models is unsolvable at all for the low wave number range because of the unstable initial state of the models used. This study clarifies the role and differences played by unsolvability and instability and suggests a consistent way to solve the problem.