Let M n , n ≥ 3, be a Hadamard manifold with strictly negative sectional curvature KM ≤ −�, � > 0. Assume that M satisfies the strict convexity condition at infinity according to (18) (see also the definition below) and, additionally, that M admits a helicoidal one parameter subgroup {'t} of isometries (i.e. there exists a geodesic of M such that 't ((s)) = (t+s) for all s,t ∈ R). We then prove that, given a compact topological {'t}−shaped hypersurface in the asymptotic boundary @1M of M (that is, the orbits of the extended action of {'t} to @1M intersect at one and only one point), and given H ∈ R, |H| < √ �, there exists a complete properly embedded constant mean curvature (CMC) H hypersurface S of M such that @1S = . This result extends Theorem 1.8 of B. Guan and J. Spruck (11) to more general ambient spaces, as rank 1 symmetric spaces of noncompact type, and allows to be {'t}−shaped with respect to more general one parameter subgroup of isometries {'t} of the ambient space. For example, in H n , can be loxodromic −shaped, where loxodromic is a curve in S n 1 = @1H n that makes a con- stant angle with a family of circles connecting two points of S n 1 . A fundamental result used to prove our main theorem, which has interest on its own, is the extension of the interior gradient esti- mates for CMC Killing graphs proved in Theorem 1 of (7) to CMC graphs of Killing submersions.