We study connected branches of non-constant 2π-periodic solutions of the Hamilton equation ẋ(t) = λJ∇H(x(t)), where λ ∈ (0,+∞), H ∈ C(R ×Rn,R) and ∇H(x0) = [ A 0 0 B ] for x0 ∈ ∇H−1(0). The Hessian ∇H(x0) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given (x0, λ0). As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system ẋ(t) = J∇H(x(t)) emanating from x0. We describe also minimal periods of trajectories near x0.