About the stability of branching solutions in the problem on capillary-gravity waves in a deep spatial layer of floating fluid

Potential flows of incompressible heavy capillary floating fluid in free-dimensional layer of infinite depth with free upper boundary are determined. Asymptotics of periodical flows in spatial layer with free upper boundary close to horizontal plane $z=0$ bifurcating from the basic flow with constant velocity $V$ in $Ox$-direction are calculated. Their orbital stability relative to pertubations of the same symmetry is investigated. Methods of group-invariant bifurcation theory and group analysis of differential equations are used. Special attention is given to cases of high-dimensional ($n \ge 4$) degeneration of the linearized operator.