Within the frame of the ESA research program SETA, "Solidification along a Eutectic Path in Ternary Alloys", experiments have been performed focussing on several distinct subtopics. One of these subtopics is to study coupled growth along the univariant eutectic reaction: L -> alpha + beta. In this paper, the influence of the growth velocity v on the morphology of the solid/liquid interface is evaluated in a ternary Al-Cu-Si alloy with a composition close to the univariant eutectic groove L -> alpha(Al) + theta-Al(2)Cu. Different structural regions can be identified in terms of the stability of the solid-liquid interface (morphological stability) and the stability of the coupling (competitive growth) during unidirectional solidification as function of the solidification parameters. It is found that two-phase planar growth with a lamellar arrangement can be obtained at a sufficiently low growth rate v. The measured interlamellar spacing follows the Jackson and Hunt relationship lambda(2)v = constant. At a higher growth velocity first a destabilisation of the solid/liquid interface is observed and finally competitive growth is observed revealing primary theta-Al(2)Cu growing ahead of the eutectic interface. It is assumed that the cellular break-up is a two-step process related to the crystallography of the system. Fitting the different morphologies into one microstructure map, an extension of the coupled zone concept as has been proposed for binary alloys is necessary.