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Symmetry, singularities and integrability in complex dynamics. V: Complete symmetry groups of certain relativistic spherically symmetric systems. (English) Zbl 1017.34035

The paper is carried out in the frame of the concept of complete symmetry groups of the classical Kepler systems, introduced by J. Krause [J. Math. Phys. 35, No. 11, 5734-5748 (1994; Zbl 0816.70006)], extending to the relativistic context of Einstein’s equations describing spherically symmetric bodies with certain conformal Killing symmetries. The concept of complete symmetry groups is discussed for the nonlinear ordinary differential equation of the form \[ y^{\prime \prime \prime}+y^{\prime \prime}+yy'=0,\tag{1} \] arisen in general relativity. The authors demonstrate that it is possible to have a complete symmetry group of equation (1) which does not have sufficient Lie point symmetries for integrability and is not integrable in the sense of Painlevé. This fact provides a simple demonstration of the nonuniqueness of the complete symmetry group. See also the other parts: The first author, S. Cotsakis and G. P. Flessas [J. Nonlinear Math. Phys. 7, 445-479 (2000; Zbl 0989.34074) and J. Math. Anal. Appl. 251, 587-609 (2000; Zbl 0992.34027)], The author, S. Moyo, S. Cotsakis and R. L. Lemmer [J. Nonlinear Math. Phys. 8, 139-156 (2001; Zbl 0992.34028)], J. Miritzis, the first author and S. Cotsakis [Gravit. Cosmol. 6, (24), 282-290 (2000; Zbl 1009.83075) and the first author, S. Cotsakis and J. Miritzis [Gravit. Cosmol. 7, (28), 311-320 (2001; Zbl 1004.83059)].

MSC:

34C14 Symmetries, invariants of ordinary differential equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
34A34 Nonlinear ordinary differential equations and systems

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