  We classify the finite type (in the sense of E. Cartan theory of
prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is
the symplectic 4-dimensional space, and show that they satisfy
$\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem
of classification of graded transitive finite-dimensional Lie algebras
$\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded
transitive finite-dimensional subalgebras of the full prolongations
$\mathfrak{p}_1^{(\infty)}$ and $\mathfrak{p}_2^{(\infty)}$, where
$\mathfrak{p}_1$ and $\mathfrak{p}_2$ are the maximal parabolic subalgebras of
$\mathfrak{sp}(V)$. We then classify all such
$\mathfrak{g}\subset\mathfrak{p}_i^{(\infty)}$, $i=1,2$, under some assumptions
and describe the associated homogeneous symplectic 4-manifolds
$(M=G/K,\omega)$. We prove that any reductive homogeneous symplectic manifold
(of any dimension) admits an invariant torsion free symplectic connection,
i.e., it is a homogeneous Fedosov manifold, and give conditions for uniqueness
of the Fedosov structure. Finally, we show that any nilpotent symplectic Lie
group (of any dimension) admits a natural invariant Fedosov structure which is
Ricci-flat.
