  For each $k\geq2$, we construct two families of surfaces with constant mean
curvature $H$ for $H\in[0,1/2]$ in $\Sigma(\kappa)\times\R$ where
$\kappa+4H^2\leq0$. The surfaces are invariant under $2\pi/k$-rotations about a
vertical fiber of $\Sigma(\kappa)\times\R$, have genus zero, and a finite
number of ends. The first family generalizes the notion of $k$-noids: It has
$k$ ends, one horizontal and $k$ vertical symmetry planes. The second family is
less symmetric and has two types of ends. Each surface arises as the conjugate
(sister) surface of a minimal graph in a homogeneous 3-manifold. The domain of
the graph is non-convex in the second family. For $\kappa=-1$ the surfaces with
constant mean curvature $H$ arise from a minimal surface in
$\widetilde{\PSL}_2(\R)$ for $H\in(0,1/2)$ and in $\Nil$ for H=1/2. For H=0,
the conjugate surfaces are both minimal in a product space.
