Some eutectic stripes have been generated in a hexagonal (Zn) - single crystal. The stripes are situated periodically with the constant interstripes spacing. The eutectic structure in the stripes consists of strengthening inter-metallic compound, Zn16Ti, and (Zn) – solid solution. The rod-like irregular eutectic structure (with branches) appears at low growth rates. The regular lamellar eutectic structure is observed at middle growth rates. The regular rod-like eutectic structure exists exclusively in the stripes at some elevated growth rates. A new thermodynamic criterion is recommended. It suggests that this eutectic regular structure is the winner in a morphological competition for which the minimum entropy production is lower. A competition between the regular rod-like and the regular lamellar eutectic growth is described by means of the proposed criterion. The formation of branches within irregular eutectic structure is referred to the state of marginal stability. A continuous transitions from the marginal stability to the stationary state are confirmed by the continuous transformations of the irregular eutectic structure into the regular one.