Generation of nonlinear ordinary differential equations with given Lie algebra of point symmetries. (Erzeugung nichtlinearer gewöhnlicher Differentialgleichungen mit vorgegebener Lie-Algebra von Punktsymmetrien.) (German) Zbl 1061.34031

Campoamor-Stursberg, Rutwig

J. Lie Theory 14, No. 2, 537-541 (2004).

The goal of this paper is to show that for every \(n\geq 4\) there exists an ordinary \(n\)th-order differential equation which admits exactly \(\text{SL}(2,\mathbb{R})\) in the usual representation \[ X_1= x\cdot\partial_x,\quad X_2= \partial_x,\quad X_3= x^2\cdot\partial_x, \] as the corresponding symmetry algebra.
At first, the author presents such an ordinary differential equation (ODE) for which the above generators are symmetries, then this ODE is modified by an additional term to exclude further symmetries. The proofs are presented rather as argumentations, whereas the concrete calculations are left to the reader.

Reviewer: Günter Czichowski (Greifswald)

MSC:

34C14	Symmetries, invariants of ordinary differential equations
17B66	Lie algebras of vector fields and related (super) algebras

Keywords:

symmetry; \(n\)th-order ordinary differential equation; symmetry algebra; \(\text{SL}(2,\mathbb{R})\)