Let M be a three-dimensional Lorentz manifold and let Z be a non-parallel, closed conformal spacelike vector field. A timelike surface \(\Sigma \) in M is said to have a canonical null direction with respect to Z if the projection \(Z^{\top }\) on the tangent space of \(\Sigma \) gives a lightlike vector field. We prove that these surfaces are not minimal. We also find that they are ruled, whose rulings are lines of curvature and lightlike geodesics in M. When M is the Minkowski space \(\mathbb {R}_1^3\) and Z is a radial vector field, we proved that \(\Sigma \) is a quasi-umbilic surface and it is not flat. Subsequently, we found a condition for quasi-umbilic surfaces to be surfaces with canonical null direction. We explored the case where \(\Sigma \) lies in De Sitter space. Here, we found that \(\Sigma \) is neither of constant mean curvature nor flat. Later, we analyzed the case where \(\Sigma \) is in a warped product \(I\times _{\rho }N\) . We can assume that the surface \(\Sigma \) is the graph of a function \(f:N \rightarrow \mathbb {R}\) . Then, it has a canonical null direction with respect to \(\rho \partial _t\) if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature K of \(\Sigma \) and the Gaussian curvature \(K_N\) of N satisfy that \(K=\frac{1}{\rho ^2}K_N\) . Finally, we analyzed when \(\Sigma \) is the inverse image of a function.
