Abstract
A systematic method for investigating the existence of nontrivial generalized Lie symmetries is presented and the associated integrals of motion for nonlinear oscillator systems with three-degrees of freedom defined in terms of the Lagrangian by L= (1)/(2) (ẋ2+ẏ2+ż2)−V(x,y,z) are constructed. Then the method is applied to study the integrability properties of quartically and cubically coupled nonlinear oscillators with three degrees of freedom. Compatibility with the Painlevé property is also investigated.
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