Poisson map

Definition.
Given two Poisson manifolds $M$ and $N$, a map $\varphi :M\to N$ is a Poisson map if

for all smooth functions on $N$.
$◼$

If the manifolds happen to be symplectic manifolds then $\varphi$ is a symplectomorphism. In particular, if the manifolds are the phase space of a classical mechanical system then $\varphi$ is a canonical map.

The Hamiltonian vector fields give rise to a local group of transformations which are Poisson maps (@olver86 proposition 6.16).