In this study, firstly we give the weighted curvatures of non-null planar curves in Lorentz-Minkowski space with density eax(2)+by(2) and obtain the planar curves whose weighted curvatures vanish in this space under the condition that the constants a and b are not zero at the same time. After giving the Frenet vectors of the non-null planar curves with zero weighted curvature in Lorentz-Minkowski space with density eax(2), we create the Smarandache curves of them. With the aid of these curves and their Smarandache curves, we get the ruled surfaces whose base curves are non-null curves of which vanishing weighted curvature and ruling curves are Smarandache curves of them. Followingly, we give some characterizations for these ruled surfaces by obtaining the mean and Gaussian curvatures, distribution parameters and striction curves of them. Also, rotational surfaces which are generated by non-null planar curves with zero weighted curvatures in Lorentz-Minkowski space E-1(3) with density eax(2) +by(2) are studied under the condition that the constants a and b are not zero at the same time. We draw the graphics of the obtained surfaces.