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Journal of Lie Theory, Vol. 10, No. 2, pp. 411-433 (2000)
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Canonical form transformation of second order differential equations with Lie symmetries

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Fritz Schwarz

GMD

Institut SCAI

53754 Sankt Augustin

Germany

fritz.schwarz@gmd.de

**Abstract:** The most difficult part of solving an ordinary differential equation (ode) by Lie's symmetry theory consists of transforming it into a canonical form corresponding to its symmetry type. In this article, for all possible symmetry types of a quasilinear second order ode, theorems are obtained that reduce the transformation into canonical form to solving linear partial differential equations (pde's) or certain Riccati equations. They allow it to determine {\eightit algorithmically} the finite transformation functions to canonical form that are Liouvillian over the base field of the given ode. The knowledge of the infinitesimal symmetry generators is {\eightit not} required. Fundamental new concepts that are applied are the Janet base of a system of linear pde's and its decomposition into completely reducible components, i. e. the analogue to Loewy's decomposition of linear ode's.

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