Homogeneous symplectic $4$-manifolds and finite dimensional Lie algebras of symplectic vector fields on the symplectic $4$-space

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 $4$-dimensional homogeneous symplectic 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 the 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.