  The minimal Lorentzian surfaces in $\mathbb{R}^4_2$ whose first normal space
is two-dimensional and whose Gauss curvature $K$ and normal curvature
$\varkappa$ satisfy $K^2-\varkappa^2 >0$ are called minimal Lorentzian surfaces
of general type. These surfaces admit canonical parameters and with respect to
such parameters are determined uniquely up to a motion in $\mathbb{R}^4_2$ by
the curvatures $K$ and $\varkappa$ satisfying a system of two natural PDEs. In
the present paper we study minimal Lorentzian surfaces in $\mathbb{R}^4_2$ and
find a Weierstrass representation with respect to isothermal parameters of any
minimal surface with two-dimensional first normal space. We also obtain a
Weierstrass representation with respect to canonical parameters of any minimal
Lorentzian surface of general type and solve explicitly the system of natural
PDEs expressing any solution to this system by means of four real functions of
one variable.
