Degenerate bifurcation of periodic solutions to an ODE caused by symmetry

We consider the equation of forced oscillations of mathematical pendulum with an external sinusoidal force. The equation has a number of symmetries, one of which is linked to a degeneracy in families of its periodic solutions which has an abnormally high codimension. The use of variational equations of higher order allows giving a strict proof of existence of such degeneracy as well as to obtain simple computational formulas, which are realized as computer algebra procedures.