Steady flows generated by the Rayleigh-Taylor instability (RTI) are considered for incompressible inviscid fluid. There is a family of steady solutions, and for the first time the problem of the solutions' stability is studied theoretically for 2D and 3D flows. The region of stable solutions is found to be very narrow and bounded by Hopf bifurcations. The influence of flow symmetry and discontinuity of dimensional crossover in RTI are shown; agreement with existing experimental and numerical data is good. Bubbles dynamics is discussed. [S0031-9007(98)06179-1].