Periodic orbit for certain Hamiltonian on the tangent bundle

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?