On some generalizations of the factorization method (Q1280832)

The classical scheme of the factorization method makes it possible to integrate a system of ordinary differential equations of the following special form: \(U_t= [U_+,U]\), \(U(0)= U_0\). Here \(U(t)\) is a function with its range in Lie algebra \({\mathfrak M}\) that can be represented as the direct sum of the vector space \({\mathfrak M}_+\) and \({\mathfrak M}_-\), each of which is a subalgebra of \({\mathfrak M}\), and \(U_+\) denotes the projection of \(U\) on \({\mathfrak M}_+\). The authors generalize this method to the case \({\mathfrak M}_+\cap{\mathfrak M}_-\neq\{0\}\). The corresponding quadratic systems are reducible to a linear system with variable coefficients. Under some additional condition it is shown by the authors that the generalized version of the factorization method can also be applied to Liouville equation-type systems of partial differential equations.