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.