We developed analytical models of the buckling failure of initially rectilinear pipeline in the environment. We specify the drawbacks of available analytical methods, which take no account of the possible longitudinal displacement of the pipeline points or the limited length of the pipeline portion, as well as, in the limiting case, fail to describe the experimental results of buckling failure in the air. The proposed models make it possible to provide analytical evaluation of the critical load, length and shape of the deformed pipeline in cases of its horizontal and vertical transversal deformations with the account of the above factors. The results obtained are corroborated by the numerical simulation of the deformation process, based on the procedure described by the authors in Part 1, as well as by the full-scale test results.