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?