The Rayleigh-Taylor instability is studied for an incompressible inviscid fluid of infinite depth for three-dimensional (3D) spatially periodic flow. The problem is formulated in terms of general conditions that allow one to find the symmetry of the observable steady structures. Analytical steady solutions for a hexagonal type of flow symmetry (plane group p6mm) are found in few orders of approximations. Interrelations between the results with various types of flow symmetry are established. Comparisons with previously studied 3D flows with "square" and "rectangular" symmetries are given. [S1063-651X(98)14312-X].