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Zbl 1036.58004
Juráš, Martin
Variational symmetries and Lie reduction for Frobenius systems of even rank. (English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--15, 2002. Sofia: Coral Press Scientific Publishing. 178-192 (2003). ISBN 954-90618-4-1/pbk

Let ${\cal I}$ be a Frobenius system (i.e., a completely integrable Pfaffian system), $\pi\in{\cal I}\wedge{\cal I}$ a closed two-form of maximal possible rank. The author deals with relations between infinitesimal symmetries $X$ of $\pi$ (defined by the property ${\cal L}_X\pi= 0$) and certain reductions of the system ${\cal I}$. In more detail, the system $\omega_1=\cdots= \omega_{r+s}= 0$ is called reducible to the system $\omega_1=\cdots= \omega_r= 0$ if $d\omega_i= 0\pmod{\omega_1,\dots,\omega_{i-1}}$ for all $r+ 1\le i\le r+s$.\par Theorems. Let ${\germ g}$ be a solvable Lie algebra of infinitesimal symmetries of ${\cal I}$. ${\cal I}$ is reducible to the Frobenius system ${\cal I}({\germ g})= \{\omega\in{\cal I}:\omega(X)= 0$ for all $X\in{\germ g}\}$ and if ${\cal I}({\germ g})= 0$, then ${\cal I}$ is solvable by quadratures. If ${\cal I}$ is a rank $2k$ Frobenius system and $X$ an infinitesimal symmetry of $\pi$, then $X$ is infinitesimal symmetry of ${\cal I}$. A one-to-one correspondence exists between certain equivalence classes of infinitesimal symmetries of $\pi$ and equivalence classes of conservation laws of ${\cal I}$.
[Jan Chrastina (Brno)]

MSC 2000:: *58A15 Exterior differential systems (Cartan theory)
58A17 Pfaffian systems
34C14 Symmetries, invariants
34A26 Geometric methods in differential equations

Keywords: Frobenius system; infinitesimal symmetry; solvable Lie group

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