On the properties of the solutions of a multidimeral system of differential equations in a complex area

This article considers a multidimensional system of differential equations with complex variables. Compiling systems of the Poincare-Lyapunov type, the characteristic equation is solved and a negative root is found. With the correct approximation of the solution of this system to a singular point with ordinary coefficients, they will be solutions of a certain system of equations of the Poincare-Lyapunov type, for which a singular point, the so-called zero point, is considered. Based on the Lyapunov theorem, this system has analytic solutions without arbitrary constants. The characteristic equation of the Poincare-Lyapunov system always has a negative root. For analytic solutions without arbitrary constants, one can always find two families of partial solutions, and they are additional properties. The condition when there are no purely imaginary roots is considered. And also, a theorem is proved that does not have regular non-spiral singular points belonging to the considered family. To determine the function, the following formulas for roots with a positive real part are obtained. In the article, using the last theorem, it is proved that if among the roots of the characteristic equation of the corresponding Poincare-Lyapunov system there are positive and negative real parts, then the system has one and only one solution. We also consider the case when there are purely imaginary roots. In the absence of zero roots, purely imaginary roots are found.