In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations for which one eigenvalue of the matrix of the linear part is zero and the remaining eigenvalues do not belong to the imaginary axis. We study the reducibility of such systems to normal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.
The current state of methods of the solution of boundary problems of mechanics of continuous environments is characterized. It is noted that packages of applied programs applied in engineering practice are based on the methods leading to solutions of boundary problems in the form of massifs of numbers. As a shortcoming the im-possibility of a reliable assessment of an error of such decisions for the majority of complex engineering chal-lenges is noted. As the alternative is stated an essence of a fictitious canonic regions method. It is shown that its application leads to solutions of boundary problems not in the form of massifs of numbers, but in the form of the functions which are identically satisfying with to the differential equations of boundary problems. The main ad-vantage of the fictitious canonic regions method - high precision of received results and possibility of a reliable assessment of their error. The review of stages of development of the fictitious canonic regions method is executed. The review of the works devoted to its application for the solution of scientific and engineering problems is executed.