Bifurcation, symmetry, and cosymmetry in differential equations unresolved with respect to the derivative with variational branching equations

The possibility of reduction of the potential-type branching equation on the basis of the general theorem on the inheritance of group symmetry was investigated by the corresponding Lyapunov-Schmidt branching equations (BE). A relationship was established between the Lie-Ovsyannikov symmetry properties of the BEs and the Yudovich cosymmetry properties for invariant problems of dynamical bifurcation. The reducibility of the BEs was ensured by cosymmetric identities satisfied by the infinitestimal operators of the corresponding Lie algebra. A general form of the BE and its potential with symmetries of the rotation groups SO(2) was determined as a correlatory for evolution equations with derivative multiplied by a degenerate Fredholm operator.