On intrinsic rotational surfaces in the Lorentz-Minkowski space

Spacelike intrinsic rotational surfaces with constant mean curvature in the Lorentz-Minkowski space $\mathbf{E}_1^3$ have been recently investigated by Brander et al., extending the known Smyth's surfaces in Euclidean space. Assuming that the surface is intrinsic rotational with coordinates $(u,v)$ and conformal factor $\rho(u)^2$, we replace the constancy of the mean curvature with the property that the Weingarten endomorphism $A$ can be expressed as $\Phi_{-\alpha(v)}\left(\begin{array}{} \lambda_1(u)\quad 0\quad\\ 0 \quad\lambda_2(u)\end{array}\right)\Phi_{\alpha(v)}$, where $\Phi_{\alpha(v)}$ is the (Euclidean or hyperbolic) rotation of angle $\alpha(v)$ at each tangent plane and $\lambda_i$ are the principal curvatures. Under these conditions, it is proved that the mean curvature is constant and $\alpha$ is a linear function. This result also covers the case that the surface is timelike. If the mean curvature is zero, we determine all spacelike and timelike intrinsic rotational surfaces with rotational angle $\alpha$. This family of surfaces includes the spacelike and timelike Enneper surfaces.