2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143555We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in arXiv:math.DG/0102183. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane: we construct and classify the entire family of these genus-zero coplanar constant mean curvature surfaces.35 pages, 10 figures; minor revisions including one new figure; to appear in Comm. Anal. GeomDifferential Geometry53A10 (Primary) 58D10, 53C42 (Secondary)text