Factorization problem with intersection

We propose a generalization of the factorization method to the case when $\mathcal G$ is a finite-dimensional Lie algebra $\mathcal G=\mathcal G_0\oplus M\oplus N$ (direct sum of vector spaces), where $\mathcal G_0$ is a subalgebra in $\mathcal G$, $M,N$ are $\mathcal G_0$-modules, and $\mathcal G_0+M$, $\mathcal G_0+N$ are subalgebras in $\mathcal G$. In particular, our construction involves the case when $\mathcal G$ is a $\mathbb Z$-graded Lie algebra. Using this generalization, we construct certain top-like systems related to algebra $so(3,1)$. According to the general scheme, these systems can be reduced to solving systems of linear equations with variable coefficients. For these systems we find polynomial first integrals and infinitesimal symmetries.