Non-degenerate surfaces of revolution in Minkowski space that satisfy the relation $aH+bK=c$

In this work, we study spacelike and timelike surfaces of revolution in Minkowski space ${\text{e}}_1^3$ that satisfy $aH+bK=c$, where $H$ and $K$ denote the mean curvature and the Gauss curvature of the surface and $a$, $b$ and $c$ are constants. The classification depends on the causal character of the axis of revolution and in all the cases, we obtain a first integral of the equation of the generating curve of the surface.