Geometry of Lie integrability by quadratures
Abstract
In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finitedimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In the second step, we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain a much larger class of explicitly integrable systems, replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency).
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 May 2015
 DOI:
 10.1088/17518113/48/21/215206
 arXiv:
 arXiv:1409.7549
 Bibcode:
 2015JPhA...48u5206C
 Keywords:

 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 37J35;
 34A34;
 34C15;
 70H06
 EPrint:
 18 pages