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 consider a kind of Taylor series and 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

$$H(x,v)=f(x)+df(x).v +\frac{1}{2} Hess(f)(x)(v,v)$$ where $Hess(f)$ is the $2-$ linear form on $T_x M$ with the formula $$Hess(f)(x)(v,w)=g(\nabla_v \nabla f, w)$$ where $\nabla$ is the LC connection associated with the Riemannian metric. Is there an example of such a Hamiltonian with a non trivial periodic orbit?