  We investigate bifurcation of closed orbits with a fixed energy level for a
class of nearly integrable Hamiltonian systems with two degrees of freedom.
More precisely, we make a joint use of Moser invariant curve theorem and
Poincar\'e-Birkhoff fixed point theorem to prove that a periodic non-degenerate
invariant torus $\mathcal{T}$ of the unperturbed problem gives rise to
infinitely many closed orbits, bifurcating from a family of tori accumulating
onto $\mathcal{T}$. The required non-degeneracy condition, which is nothing but
a reformulation of the usual non-degeneracy condition in the isoenergetic KAM
theory, is expressed in terms of the derivative of the apsidal angle with
respect to the angular momentum: in this way, tools from the theory of
time-maps of nonlinear oscillators can be used to verify it in concrete
problems. Applications are given to perturbations of central force problems in
the plane, and to equatorial geodesic dynamics for perturbations of the
Schwarzschild metric.
