On the stability of branching solutions to the problem of surface waves at the horizontal interface of two liquids, the lower of which occupies a half-space

Potential flows of two immiscible incompressible fluids in a spatial layer with an interface close to the horizontal plane $z=0$ bifurcating from the basis flows $V_1$ and $V_2$ in Ox-direction in case the lower (the heavier) fluid occupies a half-space are considered. Their orbital stability relative to perturbations with the same symmetry is investigated. Group analysis methods in bifurcation theory under the group invariance conditions are applied.