In this work we study equivariant hypersurfaces in S(n) and H(n) which are minimal or have constant mean curvature. These hypersurfaces are described via a curve in S(2) and H(2) respectively, called the generating curve. In the equivariant case, the constant mean curvature equation reduces to an ODE on the generating curve, which can be reduced by one variable using the symmetry of the problem. It then turns out that this reduced system admits a first integral. In the spherical case, we find conditions insuring closedness of the integral curves, and we deduce the existence of compact hypersurfaces which are minimal or have constant mean curvature. We also discuss the question of embeddedness of these hypersurfaces. In the hyperbolic case, we limit ourselves to the minimal case. We observe that the curves are no longer closed and again we discuss embededdness.