It is a symplectic manifold $(M,\omega)$ in which we have a prescribed a smooth function $H$ called the Hamiltonian of the system.
In other words, a Hamiltonian system is a classical mechanical system in which the state of the system is described by a point in a symplectic manifold $(M,\omega)$, and the evolution of the system is determined by the Hamiltonian function $H:M\to \mathbb{R}$. The Hamiltonian function generates one Hamiltonian vector field $X_H$, whose flow represents the time evolution of the system. The dynamics of the system is then given by Hamilton's equations, which can be written in the form $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, where $(q,p)$ are local coordinates on $M$. The Hamiltonian function encapsulates all the information about the energy and forces of the system, and the symplectic form $\omega$ describes the underlying geometric structure that governs the evolution of the system.
It is important the notion of Hamiltonian symmetry.
A slightly more general idea of Hamiltonian system is given in @olver86 page 395. A system of first order ODEs is a Hamiltonian systems if there is a function $H$ and a matrix of functions $J$ coming from a Poisson bracket#In local coordinates such that the system can be written as
$$ \frac{dx}{dt}=J(x)\nabla H(x). $$They are then generalized in @olver86 page 408 to the case
$$ \frac{dx}{dt}=J(x)\nabla H(x,t) $$although not explicitly defined. I interpret that the vector field of the system is, in this case, $\partial_t+X_H$.
A first integral of the Hamiltonian system is not a first integral of $X_H$ but a first integral of $\partial_t+X_H$, that is, a function $P(x,t)$ such that $(\partial_t+X_H) (P)=0$, i.e.,
$$ \frac{\partial P}{\partial t}+\{P,H\}=0. $$The function $P(x,t)$ generates itself a Hamiltonian vector field $X_P$, which depends on $t$ as a parameter.
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Author of the notes: Antonio J. Pan-Collantes
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