Factorization problem with intersection

We propose a generalization of the factorization method to the case when $\mathcal{G}$ is a finite dimensional Lie algebra such that $\mathcal{G}={\mathcal{G}}_0\oplus M\oplus N$ (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, we consider the case when $\mathcal{G}$ is a $\mathbb{Z}$-graded Lie algebra. Using this generalization, we construct some top-like systems related to the algebra $so(3,1)$. According to the general scheme, these systems can be reduced to linear systems with variable coefficients. For the top-like systems first integrals and infinitesimal symmetries are found.