  We study the classification of area-stationary and stable $C^2$ regular
surfaces in the space of the rigid motions of the Minkowski plane E(1,1),
equipped with its sub-Riemannian structure. We construct examples of
area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We
also prove that there exist an infinite number of $C^2$ area-stationary
surfaces with a singular curve. Finally we show the stability of $C^2$
area-stationary surfaces foliated by sub-Riemannian geodesics.
