The study of maximal and constant mean curvature spacelike hypersurfaces in Lorentzian spacetimes has been of substantial interest, from both physical and mathematical points of view. In a series of recent papers, the reviewer, jointly with Romero and Sánchez, considered the problem of uniqueness for this type of hypersurfaces in certain ambient spacetimes. In particular, they studied the uniqueness of compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes [the reviewer, {\it A. Romero} and {\it M. Sánchez}, Gen. Relativ. Gravitation 27, 71-84 (1995;

Zbl 0908.53034); Tôhoku Math. J. 49, 337-345 (1997;

Zbl 0912.53046)], and, more generally, the uniqueness of compact spacelike hypersurfaces of constant mean curvature in spacetimes which are equipped with a conformal timelike vector field [the reviewer, {\it A. Romero} and {\it M. Sánchez}, Nonlinear Anal., Theory Methods Appl. 30, 655-661 (1997;

Zbl 0891.53050)]. In the paper under review, the author considers the study of compact spacelike hypersurfaces of constant mean curvature in spacetimes which are equipped with a closed conformal timelike vector field $X$. In that case, the distribution on the spacetime orthogonal to the field $X$ provides a foliation in the spacetime by constant mean curvature spacelike leaves. Then, the author studies under what class of conditions a spacelike hypersurface with constant mean curvature must be a leaf of the foliation. In particular, the author derives some uniqueness results for compact spacelike hypersurfaces of constant mean curvature in such spacetimes. Some of the main results of the paper under review are related to, and sometimes contained in, the previous work by the reviewer, jointly with Romero and Sánchez [Nonlinear Anal., loc. cit.]. However, the author seems not to know that previous paper, which is not cited in the references.