`In  this  question a  nontrivial   periodic  orbit  is  a  periodic  orbit  which  is not a  singular point.`

Let  $p: \mathbb{R}^n \to  \mathbb{R}$  be  a  polynomial function.  We  define  the    Hamiltonian $H$  on  $\mathbb{R}^n \times \mathbb{R}^n$ as  follows:

$$H(x,y)= \sum_{\alpha}\frac{1}{\alpha!} D^{\alpha} p(x) y^{\alpha}$$  where  $\alpha$  varies  over  all  multi integer indices  and  $x,y \in \mathbb{R}^n.$

What  can  be  said about the  dynamics  of  the  corresponding  Hamiltonian  vector  field?  **Is there  an  example  of  such a  Hamiltonian  vector  field which  possess a  non trivial periodic  orbit?**

We  observe that  the  function $\sum x_i + \sum y_i$ is  a  first integral.  Are there  some  other first integrals, independent of $\sum x_i + \sum y_i$? In particular is this Hamiltonian completely integrable?


Now  we  try to  extend this questions on an arbitrary  Riemannian  manifold. So  we cut the  series  at its  second  term.  So our  question would  be  the  following:

>Let  $(M,g)$  be  a  Riemannian  manifold  and  $f:M \to  \mathbb{R}$ be  an  smooth  map.  We  define the  following Hamiltonian on the  tangent  bundle  $TM$  with its  obvious  symplectic  structure

>$$H(x,v)=f(x)+df(x).v +\frac{1}{2} Hess(f)(x)(v,v)$$

>Is  there an  example of such  a  Hamiltonian with a non trivial  periodic  orbit?