On some generalizations of the factorization method

The classical factorization method reduces the system of differential equations $U_t=[U_+,U]$ to the problem of solving algebraic equations. Here $U(t)$ belongs to a Lie algebra $\mathfrak G$ which is the direct sum of subalgebras $\mathfrak G_+$ and $\mathfrak G_-$, where “+” denotes the projection on $\mathfrak G_+$. This method is generalized to the case $\mathfrak G_+\cap\mathfrak G_-\ne\{0\}$. The corresponding quadratic systems are reduced to linear systems with varying coefficients. It is shown that the generalized version of the factorization method is also applicable to systems of partial differential equations of the Liouville type equation.