On some generalizations of the factorization method

The classical factorization method reduces the study of a system of ordinary differential equations Ut=[U+, U] to solving algebraic equations. Here U(t) belongs to a Lie algebra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{G}$$ \end{document} which is the direct sum of its subalgebras\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{G}_ + $$ \end{document} and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{G}_ - $$ \end{document}, where “+” signifies the projection on\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{G}_ + $$ \end{document}. We generalize this method to the case\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{G}_ + \cap \mathfrak{G}_ - \ne \{ 0\} $$ \end{document}. The corresponding quadratic systems are reducible to a linear system with variable coefficients. It is shown that the generalized version of the factorization method can also be applied to Liouville equation-type systems of partial differential equations.