This is Olle Stormark

A general approach to systems of partial differential equations, based on ideas developed by Lie, Cartan and Vessiot. The most basic question is that of existence of local solutions, but the methods used also provide further information - in particular as regards classifications of various families of PDE systems.

Solving a PDE system is equivalent to integrating a vector field system. The salient idea for doing the latter is to exploit the Monge characteristic vector field subsystems and their first integrals. Somewhat unexpectedly this very naturally leads to the consideration of local Lie groups and Lie pseudogroups.

1. Preface
2. Introduction and summary
3. PDE systems, pfaffian systems and vector field systems
4. Cartan's local existence theorem
5. Involutivity and the prolongation theorem
6. Drach's classification, second order PDEs in one dependent variable and Monge characteristics
7. Integration of vector field systems V satisfying dim V' = dim V +1
8. Higher order contact transformations
9. Local Lie groups
10. Structural classification of 3-dimensional Lie algebras over the complex numbers
11. Lie equations and Lie vector field systems
12. Second order PDEs in one dependent and two independent variables
13. Hyberbolic PDEs with Monge system admitting 2 or 3 first integrals
14. Classification of hyperbolic Goursat equations
15. Cartan's theory of Lie pseudogroups
16. The equivalence problem
17. Parabolic PDEs for which the Monge system admits at least two first integrals
18. The equivalence problem for general 3-dimensional pfaffian systems in five variables
19. Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge