The paper is concerned with completeness property of rank one perturbations of unperturbed operators generated by special boundary value problems (BVP) for the following $2 \times 2$ system \begin{equation} L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{equation} on a finite interval assuming that a potential matrix $Q$ is summable, and $b_1 b_2^{-1} \notin \mathbb{R}$ (essentially non-Dirac type case). We assume that unperturbed operator generated by a BVP belongs to one of the following three subclasses of the class of spectral operators: (a) normal operators; (b) operators similar either to a normal or almost normal; (c) operators that meet Riesz basis property with parentheses. We show that in each of the three cases there exists (in general, non-unique) operator generated by a quasi-periodic BVP and its certain rank-one perturbations (in the resolvent sense) gene 查看全文>>