Symmetry of $SO(2)$ and $SH(2)$ in Poincaré-Andronov-Hopf bifurcation problems with potential branching equations

For differential equations in Banach spaces with degenerate Fredholm operator at the derivative the general theorem about group symmetry inheritance by the relevant Lyapounov–Schmidt branching equation allows to construct its general form on admitted group symmetry. Here this problem is solved for the cases of the presence of $SO(2)$, $O(2)$, $SH(2)$ and $H(2)$ symmetries