We study a mechanical system subject to the action
of positional forces. It is assumed that, on a fixed set of
the system, the acting force is nonzero everywhere except for
the equilibrium points. We study symmetric periodic motions
(SPMs). The general theorem on the global bilateral extension of
the SPM to the boundary of the region of existence of the SPMs
is proved. The global continuation of the Lyapunov family with
the inheritance of a monotonic change in the period is given. It is
shown that when the period decreases, the family goes to infinity,
accompanied by a period tending to zero. An increase in the
period on the family occurs unlimitedly. In this case, the family
either goes to infinity, or adjoins a saddle- type equilibrium. In
this way the center and the saddle are connected by a family of
symmetric oscillations. Poincare law on the change in the nature
of equilibria extends to a system with n > 1 degrees of freedom.
All families of DNA base pair oscillations that bind equilibria
are found.