On a variant of a theorem on the full separation of a linear homogeneous differential equations system (Q2761557)

The author considers the system of differential equations NEWLINE\[NEWLINEdx_{j}/dt= \lambda_{j}(t,\varepsilon)x_{j}+ \sum_{k=1}^{n}b_{jk}(t,\varepsilon,\theta(t,\varepsilon))x_{k},\quad j=1,\ldots,n,NEWLINE\]NEWLINE and proves that the transformation \(x_{j}=y_{j}+\sum_{k=1(k\neq j) }^{n}q_{jk}(t,\varepsilon,\theta(t,\varepsilon))y_{k}\), \(j=1,\ldots,n\), reduces the considered system to the diagonal form NEWLINE\[NEWLINEdy_{j}/dt= \left(b_{jj}(t,\varepsilon,\theta)+ \lambda_{j}(t,\varepsilon)+ \sum_{s=1 (s\neq j)}^{n} b_{js}(t,\varepsilon,\theta)q_{sj}(t,\varepsilon,\theta)\right)y_{j}, \quad j=1,\ldots,n,NEWLINE\]NEWLINE where the \(q_{jk}\) are searched as \(q_{jk}=\sum_{m=-\infty}^{+\infty}q_{jkm}(t,\varepsilon) \exp(\operatorname {im}\theta(t,\varepsilon))\), \(j,k=1,\ldots,n\), \(j\neq k\).