Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of non-constant {$2\pi$-pe}riodic solutions of the Hamilton equation \begin{displaymath} \dot{x}(t)=\lambda J\nabla H(x(t)), \end{displaymath} where $\lambda \in \text{halfline},$ $H\in C^2({\mathbb{R}}^n\times {\mathbb{R}}^n,\mathbb{R})$ and ${\displaystyle {\mathrm{\nabla }}^{2}H(x_0)=[\begin{array}{ccc}A& 00& B\end{array}]}$ for $x_0\in \mathrm{\nabla }{H}^{-1}(0)\mathrm{.}$ The Hessian ${\mathrm{\nabla }}^{2}H(x_0)$ can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given $(x_0,{\lambda }_0)\mathrm{.}$ As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system $\dot{x}(t)=J\mathrm{\nabla }H(x(t))$ emanating from $x_0\mathrm{.}$ We describe also minimal periods of trajectories near $x_0\mathrm{.}$