Two-mode branching extremals of smooth functionals with homogeneous features of the sixth order in minima points

A description of Fredholm functionals extremal distribution, bifurcating from minima points with two-dimensional degeneration and features of the sixth order is given. The main illustrating example is the problem of heterogeneous crystal ferroelectric phases branching (based on helical model). We use modified Lyapunov–Schmidt method (reduction to key function on $\mathbb R^n$), equipped with the elements of singularities theory of smooth functions. Emphasis is put on key function with square symmetry.