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Variational symmetries and Lie reduction for Frobenius systems of even rank. (English) Zbl 1036.58004
Mladenov, Ivaïlo 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 (ISBN 954-90618-4-1/pbk). 178-192 (2003).

Let be a Frobenius system (i.e., a completely integrable Pfaffian system), π a closed two-form of maximal possible rank. The author deals with relations between infinitesimal symmetries X of π (defined by the property X π=0) and certain reductions of the system . In more detail, the system ω 1 ==ω r+s =0 is called reducible to the system ω 1 ==ω r =0 if dω i =0(modω 1 ,,ω i-1 ) for all r+1ir+s.

Theorems. Let 𝔤 be a solvable Lie algebra of infinitesimal symmetries of . is reducible to the Frobenius system (𝔤)={ω:ω(X)=0 for all X𝔤} and if (𝔤)=0, then is solvable by quadratures. If is a rank 2k Frobenius system and X an infinitesimal symmetry of π, then X is infinitesimal symmetry of . A one-to-one correspondence exists between certain equivalence classes of infinitesimal symmetries of π and equivalence classes of conservation laws of .

MSC:
58A15Exterior differential systems (Cartan theory)
58A17Pfaffian systems (global analysis)
34C14Symmetries, invariants (ODE)
34A26Geometric methods in differential equations