Reference for a type of "multi-hamiltonian" system

Let $H_1,H_2inmathcal{C}^1(mathbb{R}^3;mathbb{R})$ be two scalar fields. Consider a trajectory $vec{x}(t)inmathbb{R}^3$ such that, for all observable $finmathcal{C}^1(mathbb{R}^3;mathbb{R})$,

$$frac{mathrm{d}}{mathrm{d}t}f(x)=detbig(nabla f,nabla H_1, nabla H_2big)=frac{partial(f,H_1,H_2)}{partialvec{x}}.$$

This dynamical system recalls a Hamiltonian system with hamiltonian $H$ on the phase space $lbrace(x,p)inmathbb{R}^2rbrace$ such that for all observable $finmathcal{C}^1(mathbb{R}^2;mathbb{R})$:

$$frac{mathrm{d}}{mathrm{d}t}f(x)=detbig(nabla f,nabla Hbig)=frac{partial(f,H)}{partial(x,p)}=frac{partial f}{partial x}frac{partial H}{partial p}-frac{partial f}{partial p}frac{partial H}{partial x}=biglbrace f,Hbigrbrace,$$

the Poisson bracket. Hence I would like to say that my dynamical system is a kind of "multi-hamiltonian" system. Is there any reference in which this kind of generalisation is studied?

Edit: it can be generalised to a system with $d-1$ scalar fields $(H_i)$ on $mathbb{R}^d$ satisfying:

$$frac{mathrm{d}}{mathrm{d}t}f(x)=detbig(nabla f,nabla H_1,… nabla H_{d-1}big)=frac{partial(f,H_1,…,H_{d-1})}{partialvec{x}}.$$