Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.

**Question:** Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson bracket $\{.,.\}_t$ on $M$ such that the following holds?
$$
\{H(.,t),f\}=\{H_0,f\}_t,\quad\forall t\in\mathbb{R},\quad\forall f\in C^\infty(M,\mathbb{R}).
$$
An essentially equivalent formulation. Let $(M,\omega)$ be a smooth symplectic manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.

**Question:** Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and a smooth parameter-dependent symplectic form $\omega_t$ on $M$ such that the following holds?
$$
X^\omega_{H(.,t)}=X^{\omega_t}_{H_0},
$$
Here $X$ denotes the Hamiltonian vector field.

**Discussion:** Basically, I am trying to throw the time dependence from the Hamiltonian to the Poisson structure/symplectic form. It seems to be doable, but I am not completely sure. So instead of reinventing the wheel, I am wondering if someone knows a good reference.

If the answer is negative in this generality, I would appreciate some (hopefully mild) sufficient conditions rather than a pathological counterexample.

A reference would be more helpful than a spelled out proof. Thank you.

**Addendum:** The answer should be negative in 2 dimensions, since the symplectic form is essentially unique. But in higher dimensions there are plenty of symplectic forms.