Characterizing "non-Hamiltonian" symmetries on Poisson manifolds

[Alfred Einstead]

Let P be a Poisson manifold of dimension m, and n the maximum
dimension of its symplectic leaves. Assume that there exists m - n
invariants whose level surfaces contain the symplectic leaves.

A situation may arise where the set of symmetries that leave these
invariants fixed is strictly larger than the set of Hamiltonian fields
on the manifold P. The additional fields are what I term "non-
Hamiltonian" symmetries.

The emergence of non-Hamiltonian symmetries seems to be a situation
similar to what leads to consideration of momentum maps, but I've
never seen any reference to this phenomenon before.

Is there a name for it, and some kind of characterization theorem
which distinguishes Poisson manifolds with these extra symmetries from
those that don't have them?