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Does a constant of motion always imply hamiltonian?

If a dynamical system has a constant of motion, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system? Edit: the constant of ...
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Maximal Positive Invariant Set — Some fine print

I would like to share something I noticed on the definition of Maximal Positively Invariant Sets. Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in ...
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Invariant set: sufficient condition

Given a system $\dot x = f(x)$, $x \in \mathbb{R}^n$ with a smooth $f(x)$. Let $D$ be a set in $\mathbb{R}^n$ with a smooth boundary $\partial D$ such that $\left.\langle f(x), n(x) \rangle ...