Riesz basicity with parentheses for Dirac system with summable potential

We deal with the Dirac operator ${\mathcal{L}}_{P,U}$ generated in the space $\mathbb{H}=(L_2[0,\pi ]{)}^2$ by differential expression \begin{gather*} \ell_P(\mathbf y)=B\mathbf y’+P\mathbf y,\quad B = \begin{pmatrix} -i & 0 \ 0 & i \end{pmatrix}, \qquad P(x) = \begin{pmatrix} p_1(x) & p_2(x) \ p_3(x) & p_4(x) \end{pmatrix}, \qquad \mathbf y(x)=\begin{pmatrix}y_1(x)\ y_2(x)\end{pmatrix}, \end{gather*} and regular boundary conditions $$U(\mathbf{y})=\left(\begin{array}{cc}{u}_{11}& u_{12}\\ u_{21}& u_{22}\end{array}\right)\left(\begin{array}{c}{y}_1(0)\\ y_2(0)\end{array}\right)+\left(\begin{array}{cc}{u}_{13}& u_{14}\\ u_{23}& u_{24}\end{array}\right)\left(\begin{array}{c}{y}_1(\pi )\\ y_2(\pi )\end{array}\right)=0.$$ The entries of a matrix $P$ suppose to be summable on the segment $[0,\pi ]$ complex-valued functions. It is proved, that the operator ${\mathcal{L}}_{P,U}$ has purely discrete spectrum $\{{\lambda }_n{\}}_{n\in \mathbb{Z}}$ and ${\lambda }_n={\lambda }_n^0+o(1)$ as $\mathrm{\mid }n\mathrm{\mid }\to \mathrm{\infty }$. Here $\{{\lambda }_n^0{\}}_{n\in \mathbb{Z}}$ be the spectrum of operator $\mathcal L_{0,U}$ with zero potential and the same boundary conditions. In case this boundary conditions are strictly regular the spectrum of ${\mathcal{L}}_{P,U}$ is asymptotically simple. In this case the system of eigen and associated functions of operator ${\mathcal{L}}_{P,U}$ forms Riesz basis in $\mathbb{H}$. In case of regular but not strictly regular boundary conditions all eigenvalues of the operator ${\mathcal{L}}_{0,U}$ have multiplicity equal to $2$. In this case we give full proof of Riesz basicity of corresponding two-dimensional root subspaces of the operator ${\mathcal{L}}_{0,U}$.