The results obtained by the author in 17] are extended. An algorithm is given for the construction of an asymptotic special solution of the system of linear differential equations (1) in the case when one of the roots of the
characteristic equation possesses constant $K$-multiplicity, and the external frequency $iv(\tau)$ becomes equal to it at isolated paints of the segment
$0 \leq \tau = \varepsilon t \leq L$, where $\varepsilon$ is a small real parameter.
Citation Example:ShkiI N. I. Asymptotic behaviour of linear systems in the case of multiple roots of a characteristic equation // Ukr. Mat. Zh. - 1962. - 14, № 4. - pp. 383-392.