A new numerical-analytic method of successive approximations is suggested. It allows, for system of nonlinear ordinary differential equations $dx$/$dt = A(t)x + f(t, x)$ considered with boundary conditions $A_1x(0) - A_2x(T) = 0$ in a critical case, to study the problems of existence and approximate construction of the solutions. The matrices $A_1, A_2$ can be singular.

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