Complicated expansions of solutions to an ODE system

We consider a system of ordinary differential equations of a very general form. Let its truncated system have a solution in the form of products of powers of the independent variable and of series of powers of its multiple logarithms. We show, that under absence of critical numbers such a nonpower asymptotic behavior of a solution to the initial system can be prolonged as a power-logarithmic expansion of a solution to the initial system. It consists of series of powers of the independent variable, coefficients of which are series of powers of its multiple logarithms. We give examples of the calculations. The main attention is given to explanations of the computational algorithms.