  The paper is concerned with the Riesz basis property of a boundary value
problem associated in $L^2[0,1] \otimes \mathbb{C}^2$ with the following $2
\times 2$ Dirac type equation $$ L y = -i B^{-1} y' + Q(x) y = \lambda y, \quad
B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix}
y_1 \\ y_2 \end{pmatrix}, \quad (1) $$ with a summable potential matrix $Q \in
L^1[0,1] \otimes \mathbb{C}^{2 \times 2}$ and $b_1 < 0 < b_2$. If $b_2 = -b_1
=1$ this equation is equivalent to one dimensional Dirac equation. It is proved
that the system of root functions of a linear boundary value problem
constitutes a Riesz basis in $L^2[0,1] \otimes \mathbb{C}^2$ provided that the
boundary conditions are strictly regular.
  By analogy with the case of ordinary differential equations, boundary
conditions are called strictly regular if the eigenvalues of the corresponding
unperturbed $(Q=0)$ operator are asymptotically simple and separated. As
distinguished from the Dirac case there is no simple algebraic criterion of the
strict regularity whenever $b_1 + b_2 \not = 0$. However under certain
restrictions on coefficients of the boundary linear forms we present certain
algebraic criteria of the strict regularity in the latter case. In particular,
it is shown that regular separated boundary conditions are always strictly
regular while periodic (antiperiodic) boundary conditions are strictly regular
if and only if $b_1 + b_2 \not = 0.$
  The proof of the main result is based on existence of triangular
transformation operators for system (1). Their existence is also established
here in the case of a summable $Q$. In the case of regular (but not strictly
regular) boundary conditions we prove the Riesz basis property with
parentheses. The main results are applied to establish the Riesz basis property
of the dynamic generator of spatially non-homogenous damped Timoshenko beam
model.
