Do there exist results towards answering the following question?

Consider the Poisson algebra of regular functions $A=\mathbb R[V]$ on the symplectic vector space $V:=T^* \mathbb R^n$. Using canonical coordinates we can write $A=\mathbb R[q^1,\dots,q^n,p_1,\dots,p_n]$ with Poisson bracket being $\{q^i,p_j\}=\delta_j^i$ (all other brackets between coordinates being zero).

Question: What are the finite dimensional Lie subalgebras of the Poisson algebra $(A,\cdot,\{\:,\:\})$?

Partial answer: 1. There are subalgebras in degree 2 coming from moment maps of cotangent lifted representations. 2. There are a couple of Heisenberg subalgebras. 3. There are plenty of abelian subalgebras.

I wonder if one can give a complete classification using these building blocks.