Solvable Algebras and Integrable Systems

This paper discusses a range of questions concerning the application of solvable Lie algebras of vector fields to exact integration of systems of ordinary differential equations. The set of \(n\) independent vector fields generating a solvable Lie algebra in \(n\)-dimensional space is locally reduced to some “canonical” form. This reduction is performed constructively (using quadratures), which, in particular, allows a simultaneous integration of \(n\) systems of differential equations that are generated by these fields. Generalized completely integrable systems are introduced and their properties are investigated. General ideas are applied to integration of the Hamiltonian systems of differential equations.