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The paper is concerned with completeness property of rank one perturbationsof unperturbed operators generated by special boundary value problems (BVP) forthe 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 assumethat unperturbed operator generated by a BVP belongs to one of the followingthree 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-oneperturbations (in the resolvent sense) generated by special BVPs which arecomplete while their adjoint are not. In connection with the case (b) weinvestigate Riesz basis property of quasi-periodic BVP under certainassumptions on a potential matrix $Q$. We also find a simple formula for therank of the resolvent difference for operators corresponding to two BVPs for $n\times n$ system in terms of the coefficients of boundary linear forms.

Completeness property of one-dimensional perturbations of normal and spectral operators generated by first order systems