  The paper is concerned with the completeness property of the system of root
vectors of a boundary value problem for the following $2 \times 2$ Dirac type
equation $$ L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad y= {\rm col}(y_1,
y_2), \quad x \in [0,1], $$ $$ B = {\rm diag}(b_1, b_2), \quad b_1 < 0 < b_2,
\quad\text{and}\quad Q \in W_1^n[0,1] \otimes \mathbb{C}^{2 \times 2}, $$
subject to general non-regular two-point boundary conditions $C y(0) + D y(1) =
0$. If $b_2 = -b_1 = 1$ this equation is equivalent to the one dimensional
Dirac equation. We establish asymptotic expansion of the characteristic
determinant of this boundary value problem. This expansion directly yields new
completeness result for the system of root vectors of such boundary value
problem with non-regular and even degenerate boundary conditions. We also
present several explicit completeness results in terms of the values
$Q^{(j)}(0)$ and $Q^{(j)}(1)$. In the case of degenerate boundary conditions
and analytic $Q(\cdot)$, the criterion of completeness property is established.
