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On Lie’s symmetries for planar polynomial differential systems. (English) Zbl 0987.34027

A function \(V(x,y)\) is called an inverse integrating factor for the planar system \[ \dot x=P(x,y), \quad\dot y=Q(x,y), \] if \[ P{\partial V \over \partial x}+Q{\partial V\over\partial y}=(P_x+Q_y) \cdot V \] holds. The authors investigate polynomial planar systems and give results concerning connections between the existence of polynomial inverse integrating factors, polynomial first integrals and polynomial symmetry generators.
The results are then extended to the case of rational first integrals and symmetry generators. Some examples are given.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
22E05 Local Lie groups
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