Introduction. There exists a rather large class of different mathematical models for describing the nonlinear behavior of elastic thin-walled structures. Among these are, first of all, the equations of the theory of rods, plates, and shells. Under static loads Lagrange's principle makes it possible to obtain the equations of equilibrium for such structures and the corresponding boundary conditions. Here the equations of equilibrium are usually nonlinear partial differential equations.Independently of the methods of solving the resulting boundary-value problems, such as the methods of shooting and orthogonal sweep, Rayleigh-Ritz and Bubnov, collocations and least squares, finite differences, finite and boundary elements, and so forth, the problem of determining the stress-strain state of the structure, in the final analysis, reduces to the need to solve systems of nonlinear algebraic equations. Their solutions for various values of the load are represented by points in a space of generalized variables. Here the geometric locus of these points is a continuous curve called the load trajectory (see the figure). Due to the nonlinear behavior of the structures the load trajectories are complicated curves, sometimes loop-shaped and consisting of several curves that may intersect. Two types of singular points may be present on them. The first type consists of limit points, at which stability is lost due to a jump in the deflection: cracking (see the points A, B, C, D, E, and F in the figure). The second type consists of bifurcation points: points I and H in the figure. They are points of intersection of the principal branch of the trajectory and its bifurcation branches. At these points there is Eulerian loss of stability. At all singular points the Jacobian of the system is degenerate, and the application of the traditional methods based on different recursive schemes is impossible.Nowadays the occurrence of limit points on a load trajectory is no hindrance to the application of the well-thought-out and algorithmized methods of extension [2-7, 10, 13], which implement the idea of equivalence of arguments. However, when there are bifurcation points on the load trajectory and it is necessary to pass through them, there methods are unusable, and it becomes necessary to apply methods that differ from those just mentioned, methods that divide into two basic groups.