## On the Lie and Cartan Theory of Invariant Differential Systems

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 2, Page 229--314.

Kumpera, A.
On the Lie and Cartan Theory of Invariant Differential Systems
Given a $g$-invariant differential system $S$, we first construct the auxiliary system $\mathcal{S}$ whose \textit{form\/} only depends upon the structure of the Lie algebra $g$ and upon a specific choice of an infinitesimal action of this algebra. Next, we discuss the reduction procedure associated to a Jordan-HÃ¶lder sequence in $g$. This results in a finite family $S_i$ of differential systems invariant under simple algebras and the integration problem for $S$ can be replaced by the integration problems for the auxiliary systems $\mathcal{S}_i$. The theory is illustrated in the case of solvable algebras, the classical algebras $sl(n,\mathbf{R})$ and $so(n,\mathbf{R})$ and the real forms of $g_2$. It is also applied to systems of partial differential equations of finite type and to the integration of the characteristics of involutive systems of two second order equations and single second order equations with integrable double Monge characteristics. Some space is also devoted to integrating factors and Jacobi multipliers in connection with infinitesimal automorphisms.