Fundamental vector field

In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.
Motivation.
Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if formula_1 is a smooth manifold and formula_2 is a smooth vector field, one is interested in finding integral curves to formula_3. More precisely, given formula_4 one is interested in curves formula_5 such that
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If formula_3 is furthermore a complete vector field, then the flow of formula_3, defined as the collection of all integral curves for formula_3, is a diffeomorphism of formula_10. The flow formula_11 given by formula_12 is in fact an action of the additive Lie group formula_13 on formula_10.
Conversely, every smooth action formula_15 defines a complete vector field formula_3 via the equation
It is then a simple result that there is a bijective correspondence between formula_18 actions on formula_1 and complete vector fields on formula_1.
In the language of flow theory, the vector field formula_3 is called the "infinitesimal generator". Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on formula_1.
Definition.
Let formula_23 be a Lie group with corresponding Lie algebra formula_24. Furthermore, let formula_1 be a smooth manifold endowed with a smooth action formula_26. Denote the map formula_27 such that formula_28, called the "orbit map of formula_29 corresponding to formula_30". For formula_31, the fundamental vector field formula_32 corresponding to formula_3 is any of the following equivalent definitions:
where formula_37 is the differential of a smooth map and formula_38 is the zero vector in the vector space formula_39.
The map formula_40 can then be shown to be a Lie algebra homomorphism.
Applications.
Lie groups.
The Lie algebra of a Lie group formula_23 may be identified with either the left- or right-invariant vector fields on formula_23. It is a well known result that such vector fields are isomorphic to formula_43, the tangent space at identity. In fact, if we let formula_23 act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.
Hamiltonian group actions.
In the motivation, it was shown that there is a bijective correspondence between smooth formula_18 actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields.
A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold formula_46, we say that formula_47 is a Hamiltonian vector field if there exists a smooth function formula_48 satisfying
where the map formula_50 is the interior product. This motivatives the definition of a "Hamiltonian group action" as follows: If formula_23 is a Lie group with Lie algebra formula_24 and formula_53 is a group action of formula_23 on a smooth manifold formula_1, then we say that formula_56 is a Hamiltonian group action if there exists a moment map formula_57 such that for each formula_31, 
where formula_60 and formula_32 is the fundamental vector field of formula_3