On some generalizations of the factorization method

The classical factorization method reduces the study of a system of ordinary differential equations U-t = [U+,U] to solving algebraic equations. Here U(t) belongs to a Lie algebra B which is the direct sum of its subalgebras B+ and B-, where ''+'' signifies the projection on B+. We generalize this method to the case B+ boolean AND B- not equal {0}. 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.