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We study the problem of determination of conditions for the existence of solutions to weakly nonlinear periodic boundary-value problems for systems of ordinary differential equations with switchings and construct these solutions. The critical case where the equation for generating amplitudes of weakly nonlinear periodic boundary-value problem with switchings does not turn into the identity is analyzed. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of the weakly nonlinear periodic boundary-value problem with switchings in the critical case on the basis of the generalized Newton–Kantorovich theorem. We investigate the case of a nonlinear equation whose dimension does not coincide with the dimension of the unknown in the case where the rank of Jacobian of the nonlinear equation is complete. As an example of application of the constructed iterative scheme based on the generalized Newton–Kantorovich theorem, we obtain approximations to the solutions of periodic boundary-value problem for the mathematical model of nonisothermal chemical reaction. To check the accuracy of the obtained approximations, we evaluate the discrepancies in the original equation. We establish an estimate for the length of the range of a small parameter, where the iterative procedure aimed at the construction of solutions to the weakly nonlinear periodic boundary-value problem with switchings in the mathematical model of nonisothermal chemical reaction is convergent. To this end, we use the conditions of convergence, in particular, the requirement of contraction for the operator used to obtain the required solution of the original problem under the assumption of applicability of the generalized Newton–Kantorovich theorem. [ABSTRACT FROM AUTHOR]