A circular cylindrical shell compressed from the outer side of the surface by a high-pressure fluid loses stability of its form at a certain value of the external pressure (“collapse” or nonlinear buckling occurs). As a result, bumps and dents extended along the shell generatrix appear on the shell surface; in the shell cross section, these structures alternate [1]. Deformations of the thin shell remain elastic and, as a whole, are consistent with its geometric bending [2]. Therefore, the initial stage of changes in the shell form can be described within the framework of the nonlinear theory of elasticity [3–5]. Equations of the nonlinear theory of elasticity take into account not only the geometric nonlinearity of the problem caused by nonlinearity of the strain tensor, but also the physical nonlinearity characterizing the material properties and described by the highest invariants of the strain tensor in the expansion of the expression for nonlinearly elastic energy of the medium. Correct allowance of the highest invariants is of principal importance, because the emergence of nonlinearity effects leads to localization of shell bendings. In the final analysis, formation of spatially localized patterns from dents on the shell surface at the initial stage of changes in its form is the result of interaction of nonlinearity and dispersion effects. One important feature of the problem is the absence of dispersion terms in the original equations of the nonlinear theory of elasticity. In simplified models of shells, dispersion terms appear owing to elimination of the “fast” variable characterizing the nonuniformity of strains along the normal to the shell surface. There are some novel methods [6] that allow obtaining simplified models, based on proven equations of the nonlinear theory of elasticity without using ap riorihypotheses and with controlled accuracy in terms of small parameters characterizing the shell size, magnitude of external stress, space and time scales of deformations, and geometric and physical nonlinearity of the problem. Such methods reveal latent dynamic symmetry of the problem; therefore, the simplified equations are universal and close to integrated models, which allows their solutions to be analyzed in detail by methods of the advanced theory of solitons. Nevertheless, the nonlinearly elastic dynamics of shells near their stability thresholds and the possibilities of its approximation by integrated models have not been studied yet. A variant of the reductive perturbation theory suitable for solving nonlinear boundary-value problems, where the final surface of the deformed shell is not known in advance and is found in the course of solving the problem, is proposed in the present paper. The expression for the initial nonlinearly elastic energy of the material is presented in the form of an expansion over all strain-tensor invariants admitted by medium symmetry. The method proposed allows automatic selection of invariants from equations of the nonlinear theory of elasticity and contributions of