Projections of Singular Vectors of Verma Modules over Rank 2 Kac–Moody Lie Algebras

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac–Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103–113].
In the simpler case of $\mathcal A_1^1$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154–156].